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Properties

Label 4-95e2-1.1-c4e2-0-1
Degree 44
Conductor 90259025
Sign 11
Analytic cond. 96.435296.4352
Root an. cond. 3.133713.13371
Motivic weight 44
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  + 13·4-s + 50·5-s − 142·9-s − 124·11-s − 87·16-s + 722·19-s + 650·20-s + 1.87e3·25-s − 1.84e3·36-s − 1.61e3·44-s − 7.10e3·45-s + 4.80e3·49-s − 6.20e3·55-s + 1.42e4·61-s − 4.45e3·64-s + 9.38e3·76-s − 4.35e3·80-s + 1.36e4·81-s + 3.61e4·95-s + 1.76e4·99-s + 2.43e4·100-s + 4.01e4·101-s − 1.77e4·121-s + 6.25e4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.812·4-s + 2·5-s − 1.75·9-s − 1.02·11-s − 0.339·16-s + 2·19-s + 13/8·20-s + 3·25-s − 1.42·36-s − 0.832·44-s − 3.50·45-s + 2·49-s − 2.04·55-s + 3.83·61-s − 1.08·64-s + 13/8·76-s − 0.679·80-s + 2.07·81-s + 4·95-s + 1.79·99-s + 2.43·100-s + 3.94·101-s − 1.21·121-s + 4·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯

Functional equation

Λ(s)=(9025s/2 ΓC(s)2 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(9025s/2 ΓC(s+2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 90259025    =    52⋅1925^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 96.435296.4352
Root analytic conductor: 3.133713.13371
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 9025, ( :2,2), 1)(4,\ 9025,\ (\ :2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 3.5433377983.543337798
L(12)L(\frac12) ≈\approx 3.5433377983.543337798
L(3)L(3) 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)
bad5C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
19C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
good2C22C_2^2 1−13T2+p8T4 1 - 13 T^{2} + p^{8} T^{4}
3C22C_2^2 1+142T2+p8T4 1 + 142 T^{2} + p^{8} T^{4}
7C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
11C2C_2 (1+62T+p4T2)2 ( 1 + 62 T + p^{4} T^{2} )^{2}
13C22C_2^2 1+52622T2+p8T4 1 + 52622 T^{2} + p^{8} T^{4}
17C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
23C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
29C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
31C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
37C22C_2^2 1−3237298T2+p8T4 1 - 3237298 T^{2} + p^{8} T^{4}
41C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
43C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
47C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
53C22C_2^2 1+15154382T2+p8T4 1 + 15154382 T^{2} + p^{8} T^{4}
59C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
61C2C_2 (1−7138T+p4T2)2 ( 1 - 7138 T + p^{4} T^{2} )^{2}
67C22C_2^2 1+3364622T2+p8T4 1 + 3364622 T^{2} + p^{8} T^{4}
71C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
73C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
79C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
83C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
89C1C_1×\timesC1C_1 (1−p2T)2(1+p2T)2 ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}
97C22C_2^2 1−60577618T2+p8T4 1 - 60577618 T^{2} + p^{8} 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

−13.57334943631846293688007237545, −13.18836900876708261982688968689, −12.54643651824025999483925167147, −11.67142682040378487914843403625, −11.50513753515577700174771054630, −10.82477619658826124356223081458, −10.21498510190369114511038332773, −9.898354266982772451834703022771, −9.007168983878526367304391043211, −8.848879223948621224497156625464, −7.911359142333311285752965927687, −7.19133250618896566611253659942, −6.55050977467251966660474252264, −5.83737890371271249716412607421, −5.44855585979411610412198506732, −5.06408842290981846085975709017, −3.32073381822115920946437634512, −2.53597628261584386114806763171, −2.25572542021785236939160522503, −0.882534227291798518873035730630, 0.882534227291798518873035730630, 2.25572542021785236939160522503, 2.53597628261584386114806763171, 3.32073381822115920946437634512, 5.06408842290981846085975709017, 5.44855585979411610412198506732, 5.83737890371271249716412607421, 6.55050977467251966660474252264, 7.19133250618896566611253659942, 7.911359142333311285752965927687, 8.848879223948621224497156625464, 9.007168983878526367304391043211, 9.898354266982772451834703022771, 10.21498510190369114511038332773, 10.82477619658826124356223081458, 11.50513753515577700174771054630, 11.67142682040378487914843403625, 12.54643651824025999483925167147, 13.18836900876708261982688968689, 13.57334943631846293688007237545

Graph of the ZZ-function along the critical line