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Properties

Label 4-525e2-1.1-c3e2-0-7
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 959.512959.512
Root an. cond. 5.565605.56560
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 6·3-s + 4-s + 18·6-s − 14·7-s − 3·8-s + 27·9-s + 62·11-s − 6·12-s + 6·13-s + 42·14-s + 9·16-s − 40·17-s − 81·18-s − 122·19-s + 84·21-s − 186·22-s − 16·23-s + 18·24-s − 18·26-s − 108·27-s − 14·28-s + 352·29-s + 66·31-s + 165·32-s − 372·33-s + 120·34-s + ⋯
L(s)  = 1  − 1.06·2-s − 1.15·3-s + 1/8·4-s + 1.22·6-s − 0.755·7-s − 0.132·8-s + 9-s + 1.69·11-s − 0.144·12-s + 0.128·13-s + 0.801·14-s + 9/64·16-s − 0.570·17-s − 1.06·18-s − 1.47·19-s + 0.872·21-s − 1.80·22-s − 0.145·23-s + 0.153·24-s − 0.135·26-s − 0.769·27-s − 0.0944·28-s + 2.25·29-s + 0.382·31-s + 0.911·32-s − 1.96·33-s + 0.605·34-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 959.512959.512
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 275625, ( :3/2,3/2), 1)(4,\ 275625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.84352825140.8435282514
L(12)L(\frac12) ≈\approx 0.84352825140.8435282514
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 (1+pT)2 ( 1 + p T )^{2}
5 1 1
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2D4D_{4} 1+3T+p3T2+3p3T3+p6T4 1 + 3 T + p^{3} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−62T+3582T2−62p3T3+p6T4 1 - 62 T + 3582 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−6T+3378T2−6p3T3+p6T4 1 - 6 T + 3378 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+40T+8750T2+40p3T3+p6T4 1 + 40 T + 8750 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+122T+17398T2+122p3T3+p6T4 1 + 122 T + 17398 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+16T+34pT2+16p3T3+p6T4 1 + 16 T + 34 p T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−352T+78278T2−352p3T3+p6T4 1 - 352 T + 78278 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−66T+45870T2−66p3T3+p6T4 1 - 66 T + 45870 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−188T+44542T2−188p3T3+p6T4 1 - 188 T + 44542 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−16T+18350T2−16p3T3+p6T4 1 - 16 T + 18350 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−396T+2226pT2−396p3T3+p6T4 1 - 396 T + 2226 p T^{2} - 396 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−4pT+15582T2−4p4T3+p6T4 1 - 4 p T + 15582 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4}
53D4D_{4} 1+982T+504354T2+982p3T3+p6T4 1 + 982 T + 504354 T^{2} + 982 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−516T+418118T2−516p3T3+p6T4 1 - 516 T + 418118 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+880T+575238T2+880p3T3+p6T4 1 + 880 T + 575238 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−356T+100374T2−356p3T3+p6T4 1 - 356 T + 100374 T^{2} - 356 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−310T+664038T2−310p3T3+p6T4 1 - 310 T + 664038 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+326T+789802T2+326p3T3+p6T4 1 + 326 T + 789802 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−1832T+1824478T2−1832p3T3+p6T4 1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−680T+744870T2−680p3T3+p6T4 1 - 680 T + 744870 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−796T+411158T2−796p3T3+p6T4 1 - 796 T + 411158 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−670T+1145410T2−670p3T3+p6T4 1 - 670 T + 1145410 T^{2} - 670 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−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

−10.44952857988875015793601439060, −10.40099230716399718473465553066, −9.540365068206106608860730546166, −9.514644768178388124076130362418, −8.873951856178545941469969180437, −8.778035039137146358886626658437, −7.998355039869237451727281040568, −7.66450475351796549017430363913, −6.72590460896638801413599308830, −6.46481252689732640356209446685, −6.34581754091404579682857301941, −5.94609151761960009021876136681, −4.91997094614412997784028654818, −4.47848582870770620047149031389, −4.13752717693359713465153361429, −3.36926902953674733975486515684, −2.55337736460355621813102975690, −1.73711471114374038831705788498, −0.66305326649196143557993196086, −0.66253792770971114502580459844, 0.66253792770971114502580459844, 0.66305326649196143557993196086, 1.73711471114374038831705788498, 2.55337736460355621813102975690, 3.36926902953674733975486515684, 4.13752717693359713465153361429, 4.47848582870770620047149031389, 4.91997094614412997784028654818, 5.94609151761960009021876136681, 6.34581754091404579682857301941, 6.46481252689732640356209446685, 6.72590460896638801413599308830, 7.66450475351796549017430363913, 7.998355039869237451727281040568, 8.778035039137146358886626658437, 8.873951856178545941469969180437, 9.514644768178388124076130362418, 9.540365068206106608860730546166, 10.40099230716399718473465553066, 10.44952857988875015793601439060

Graph of the ZZ-function along the critical line