Properties

Label 4-45e2-1.1-c13e2-0-2
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $2328.44$
Root an. cond. $6.94650$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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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

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

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2328.44\)
Root analytic conductor: \(6.94650\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2025,\ (\ :13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - p^{6} T )^{2} \)
good2$D_{4}$ \( 1 - 5 p^{4} T + 625 p^{4} T^{2} - 5 p^{17} T^{3} + p^{26} T^{4} \)
7$D_{4}$ \( 1 + 616300 T + 38195042750 p T^{2} + 616300 p^{13} T^{3} + p^{26} T^{4} \)
11$D_{4}$ \( 1 - 2467136 T + 1885360936226 p T^{2} - 2467136 p^{13} T^{3} + p^{26} T^{4} \)
13$D_{4}$ \( 1 - 6517860 T + 474444414946750 T^{2} - 6517860 p^{13} T^{3} + p^{26} T^{4} \)
17$D_{4}$ \( 1 + 633460 T + 11487320401034950 T^{2} + 633460 p^{13} T^{3} + p^{26} T^{4} \)
19$D_{4}$ \( 1 + 374063400 T + 6186197213067922 p T^{2} + 374063400 p^{13} T^{3} + p^{26} T^{4} \)
23$D_{4}$ \( 1 + 621982140 T + 381235795435716850 T^{2} + 621982140 p^{13} T^{3} + p^{26} T^{4} \)
29$D_{4}$ \( 1 - 7795134100 T + 33093039872067713278 T^{2} - 7795134100 p^{13} T^{3} + p^{26} T^{4} \)
31$D_{4}$ \( 1 + 4957819816 T + 26301842446694653646 T^{2} + 4957819816 p^{13} T^{3} + p^{26} T^{4} \)
37$D_{4}$ \( 1 - 21833071780 T + \)\(28\!\cdots\!50\)\( T^{2} - 21833071780 p^{13} T^{3} + p^{26} T^{4} \)
41$D_{4}$ \( 1 + 5565813644 T + \)\(18\!\cdots\!26\)\( T^{2} + 5565813644 p^{13} T^{3} + p^{26} T^{4} \)
43$D_{4}$ \( 1 + 52510877700 T + \)\(41\!\cdots\!50\)\( T^{2} + 52510877700 p^{13} T^{3} + p^{26} T^{4} \)
47$D_{4}$ \( 1 + 92855886340 T + \)\(12\!\cdots\!50\)\( T^{2} + 92855886340 p^{13} T^{3} + p^{26} T^{4} \)
53$D_{4}$ \( 1 + 266248876180 T + \)\(64\!\cdots\!50\)\( T^{2} + 266248876180 p^{13} T^{3} + p^{26} T^{4} \)
59$D_{4}$ \( 1 + 253501607800 T + \)\(16\!\cdots\!58\)\( T^{2} + 253501607800 p^{13} T^{3} + p^{26} T^{4} \)
61$D_{4}$ \( 1 + 377459239836 T + \)\(32\!\cdots\!86\)\( T^{2} + 377459239836 p^{13} T^{3} + p^{26} T^{4} \)
67$D_{4}$ \( 1 + 2375782313740 T + \)\(25\!\cdots\!50\)\( T^{2} + 2375782313740 p^{13} T^{3} + p^{26} T^{4} \)
71$D_{4}$ \( 1 - 556190102776 T + \)\(15\!\cdots\!66\)\( T^{2} - 556190102776 p^{13} T^{3} + p^{26} T^{4} \)
73$D_{4}$ \( 1 + 556465382460 T + \)\(29\!\cdots\!50\)\( T^{2} + 556465382460 p^{13} T^{3} + p^{26} T^{4} \)
79$D_{4}$ \( 1 + 2408230567600 T + \)\(44\!\cdots\!78\)\( T^{2} + 2408230567600 p^{13} T^{3} + p^{26} T^{4} \)
83$D_{4}$ \( 1 - 3898295602980 T + \)\(20\!\cdots\!50\)\( T^{2} - 3898295602980 p^{13} T^{3} + p^{26} T^{4} \)
89$D_{4}$ \( 1 + 106847675700 T + \)\(43\!\cdots\!38\)\( T^{2} + 106847675700 p^{13} T^{3} + p^{26} T^{4} \)
97$D_{4}$ \( 1 - 18744409451140 T + \)\(21\!\cdots\!50\)\( T^{2} - 18744409451140 p^{13} T^{3} + p^{26} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line