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Properties

Label 4-315e2-1.1-c3e2-0-8
Degree 44
Conductor 9922599225
Sign 11
Analytic cond. 345.424345.424
Root an. cond. 4.311104.31110
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4-s + 10·5-s + 14·7-s − 3·8-s − 30·10-s − 62·11-s − 6·13-s − 42·14-s + 9·16-s − 40·17-s − 122·19-s + 10·20-s + 186·22-s − 16·23-s + 75·25-s + 18·26-s + 14·28-s − 352·29-s + 66·31-s + 165·32-s + 120·34-s + 140·35-s − 188·37-s + 366·38-s − 30·40-s − 16·41-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s + 0.894·5-s + 0.755·7-s − 0.132·8-s − 0.948·10-s − 1.69·11-s − 0.128·13-s − 0.801·14-s + 9/64·16-s − 0.570·17-s − 1.47·19-s + 0.111·20-s + 1.80·22-s − 0.145·23-s + 3/5·25-s + 0.135·26-s + 0.0944·28-s − 2.25·29-s + 0.382·31-s + 0.911·32-s + 0.605·34-s + 0.676·35-s − 0.835·37-s + 1.56·38-s − 0.118·40-s − 0.0609·41-s + ⋯

Functional equation

Λ(s)=(99225s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(99225s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9922599225    =    34⋅52⋅723^{4} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 345.424345.424
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 99225, ( :3/2,3/2), 1)(4,\ 99225,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
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

−10.73367796985506401003902973602, −10.71628834787991071714588334763, −9.915843558118206884024008710122, −9.589166291319292280483701314920, −9.088519870035481934270409011195, −8.749022061685460881585497054662, −8.112804230021044235271879989329, −7.919320767137666454790690186965, −7.34645259867073888196785873983, −6.57278425894894481640026427981, −6.11845432029063796772736407004, −5.57744502018219020086240826481, −4.79606338446240584283964420960, −4.71277872264457071646378309301, −3.57332554106405253211400630797, −2.72285383104770266038427571937, −2.06831059306442767213738978273, −1.56122235597130864247503891068, 0, 0, 1.56122235597130864247503891068, 2.06831059306442767213738978273, 2.72285383104770266038427571937, 3.57332554106405253211400630797, 4.71277872264457071646378309301, 4.79606338446240584283964420960, 5.57744502018219020086240826481, 6.11845432029063796772736407004, 6.57278425894894481640026427981, 7.34645259867073888196785873983, 7.919320767137666454790690186965, 8.112804230021044235271879989329, 8.749022061685460881585497054662, 9.088519870035481934270409011195, 9.589166291319292280483701314920, 9.915843558118206884024008710122, 10.71628834787991071714588334763, 10.73367796985506401003902973602

Graph of the ZZ-function along the critical line