Properties

Label 4-45e2-1.1-c13e2-0-2
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 2328.442328.44
Root an. cond. 6.946506.94650
Motivic weight 1313
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 80·2-s − 3.60e3·4-s + 3.12e4·5-s − 6.16e5·7-s − 4.32e5·8-s + 2.50e6·10-s + 2.46e6·11-s + 6.51e6·13-s − 4.93e7·14-s − 1.32e7·16-s − 6.33e5·17-s − 3.74e8·19-s − 1.12e8·20-s + 1.97e8·22-s − 6.21e8·23-s + 7.32e8·25-s + 5.21e8·26-s + 2.21e9·28-s + 7.79e9·29-s − 4.95e9·31-s − 4.46e9·32-s − 5.06e7·34-s − 1.92e10·35-s + 2.18e10·37-s − 2.99e10·38-s − 1.35e10·40-s − 5.56e9·41-s + ⋯
L(s)  = 1  + 0.883·2-s − 0.439·4-s + 0.894·5-s − 1.97·7-s − 0.583·8-s + 0.790·10-s + 0.419·11-s + 0.374·13-s − 1.75·14-s − 0.198·16-s − 0.00636·17-s − 1.82·19-s − 0.393·20-s + 0.371·22-s − 0.876·23-s + 3/5·25-s + 0.331·26-s + 0.870·28-s + 2.43·29-s − 1.00·31-s − 0.735·32-s − 0.00562·34-s − 1.77·35-s + 1.39·37-s − 1.61·38-s − 0.521·40-s − 0.182·41-s + ⋯

Functional equation

Λ(s)=(2025s/2ΓC(s)2L(s)=(Λ(14s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
Λ(s)=(2025s/2ΓC(s+13/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20252025    =    34523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 2328.442328.44
Root analytic conductor: 6.946506.94650
Motivic weight: 1313
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2025, ( :13/2,13/2), 1)(4,\ 2025,\ (\ :13/2, 13/2),\ 1)

Particular Values

L(7)L(7) == 00
L(12)L(\frac12) == 00
L(152)L(\frac{15}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1 (1p6T)2 ( 1 - p^{6} T )^{2}
good2D4D_{4} 15p4T+625p4T25p17T3+p26T4 1 - 5 p^{4} T + 625 p^{4} T^{2} - 5 p^{17} T^{3} + p^{26} T^{4}
7D4D_{4} 1+616300T+38195042750pT2+616300p13T3+p26T4 1 + 616300 T + 38195042750 p T^{2} + 616300 p^{13} T^{3} + p^{26} T^{4}
11D4D_{4} 12467136T+1885360936226pT22467136p13T3+p26T4 1 - 2467136 T + 1885360936226 p T^{2} - 2467136 p^{13} T^{3} + p^{26} T^{4}
13D4D_{4} 16517860T+474444414946750T26517860p13T3+p26T4 1 - 6517860 T + 474444414946750 T^{2} - 6517860 p^{13} T^{3} + p^{26} T^{4}
17D4D_{4} 1+633460T+11487320401034950T2+633460p13T3+p26T4 1 + 633460 T + 11487320401034950 T^{2} + 633460 p^{13} T^{3} + p^{26} T^{4}
19D4D_{4} 1+374063400T+6186197213067922pT2+374063400p13T3+p26T4 1 + 374063400 T + 6186197213067922 p T^{2} + 374063400 p^{13} T^{3} + p^{26} T^{4}
23D4D_{4} 1+621982140T+381235795435716850T2+621982140p13T3+p26T4 1 + 621982140 T + 381235795435716850 T^{2} + 621982140 p^{13} T^{3} + p^{26} T^{4}
29D4D_{4} 17795134100T+33093039872067713278T27795134100p13T3+p26T4 1 - 7795134100 T + 33093039872067713278 T^{2} - 7795134100 p^{13} T^{3} + p^{26} T^{4}
31D4D_{4} 1+4957819816T+26301842446694653646T2+4957819816p13T3+p26T4 1 + 4957819816 T + 26301842446694653646 T^{2} + 4957819816 p^{13} T^{3} + p^{26} T^{4}
37D4D_{4} 121833071780T+ 1 - 21833071780 T + 28 ⁣ ⁣5028\!\cdots\!50T221833071780p13T3+p26T4 T^{2} - 21833071780 p^{13} T^{3} + p^{26} T^{4}
41D4D_{4} 1+5565813644T+ 1 + 5565813644 T + 18 ⁣ ⁣2618\!\cdots\!26T2+5565813644p13T3+p26T4 T^{2} + 5565813644 p^{13} T^{3} + p^{26} T^{4}
43D4D_{4} 1+52510877700T+ 1 + 52510877700 T + 41 ⁣ ⁣5041\!\cdots\!50T2+52510877700p13T3+p26T4 T^{2} + 52510877700 p^{13} T^{3} + p^{26} T^{4}
47D4D_{4} 1+92855886340T+ 1 + 92855886340 T + 12 ⁣ ⁣5012\!\cdots\!50T2+92855886340p13T3+p26T4 T^{2} + 92855886340 p^{13} T^{3} + p^{26} T^{4}
53D4D_{4} 1+266248876180T+ 1 + 266248876180 T + 64 ⁣ ⁣5064\!\cdots\!50T2+266248876180p13T3+p26T4 T^{2} + 266248876180 p^{13} T^{3} + p^{26} T^{4}
59D4D_{4} 1+253501607800T+ 1 + 253501607800 T + 16 ⁣ ⁣5816\!\cdots\!58T2+253501607800p13T3+p26T4 T^{2} + 253501607800 p^{13} T^{3} + p^{26} T^{4}
61D4D_{4} 1+377459239836T+ 1 + 377459239836 T + 32 ⁣ ⁣8632\!\cdots\!86T2+377459239836p13T3+p26T4 T^{2} + 377459239836 p^{13} T^{3} + p^{26} T^{4}
67D4D_{4} 1+2375782313740T+ 1 + 2375782313740 T + 25 ⁣ ⁣5025\!\cdots\!50T2+2375782313740p13T3+p26T4 T^{2} + 2375782313740 p^{13} T^{3} + p^{26} T^{4}
71D4D_{4} 1556190102776T+ 1 - 556190102776 T + 15 ⁣ ⁣6615\!\cdots\!66T2556190102776p13T3+p26T4 T^{2} - 556190102776 p^{13} T^{3} + p^{26} T^{4}
73D4D_{4} 1+556465382460T+ 1 + 556465382460 T + 29 ⁣ ⁣5029\!\cdots\!50T2+556465382460p13T3+p26T4 T^{2} + 556465382460 p^{13} T^{3} + p^{26} T^{4}
79D4D_{4} 1+2408230567600T+ 1 + 2408230567600 T + 44 ⁣ ⁣7844\!\cdots\!78T2+2408230567600p13T3+p26T4 T^{2} + 2408230567600 p^{13} T^{3} + p^{26} T^{4}
83D4D_{4} 13898295602980T+ 1 - 3898295602980 T + 20 ⁣ ⁣5020\!\cdots\!50T23898295602980p13T3+p26T4 T^{2} - 3898295602980 p^{13} T^{3} + p^{26} T^{4}
89D4D_{4} 1+106847675700T+ 1 + 106847675700 T + 43 ⁣ ⁣3843\!\cdots\!38T2+106847675700p13T3+p26T4 T^{2} + 106847675700 p^{13} T^{3} + p^{26} T^{4}
97D4D_{4} 118744409451140T+ 1 - 18744409451140 T + 21 ⁣ ⁣5021\!\cdots\!50T218744409451140p13T3+p26T4 T^{2} - 18744409451140 p^{13} T^{3} + p^{26} T^{4}
show more
show less
   L(s)=p j=14(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.73591540311249138851913966273, −12.63399231470785806222527772929, −11.78104840321772313237440450521, −10.83518279080862306124119364239, −10.07114919878083405459717323229, −9.937062189600086689558788208416, −8.954454909570612616119694337647, −8.854294366333990492379938197363, −7.74897356819101288692771710774, −6.57201332815399923541023244790, −6.22814505082278178330457576962, −6.08886496033399923850353020686, −4.74357319851420580747765435389, −4.51049237935660213249048214948, −3.48266709739154217327358336712, −3.11950391507079835013279077268, −2.17265478868093070678338155590, −1.32266173339432277477382555465, 0, 0, 1.32266173339432277477382555465, 2.17265478868093070678338155590, 3.11950391507079835013279077268, 3.48266709739154217327358336712, 4.51049237935660213249048214948, 4.74357319851420580747765435389, 6.08886496033399923850353020686, 6.22814505082278178330457576962, 6.57201332815399923541023244790, 7.74897356819101288692771710774, 8.854294366333990492379938197363, 8.954454909570612616119694337647, 9.937062189600086689558788208416, 10.07114919878083405459717323229, 10.83518279080862306124119364239, 11.78104840321772313237440450521, 12.63399231470785806222527772929, 12.73591540311249138851913966273

Graph of the ZZ-function along the critical line