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Properties

Label 4-45e2-1.1-c7e2-0-1
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 197.608197.608
Root an. cond. 3.749313.74931
Motivic weight 77
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  + 140·4-s − 150·5-s + 1.36e4·11-s + 3.21e3·16-s + 1.37e4·19-s − 2.10e4·20-s − 5.56e4·25-s − 5.11e4·29-s + 1.64e5·31-s + 1.06e6·41-s + 1.91e6·44-s + 1.47e6·49-s − 2.04e6·55-s − 2.87e6·59-s + 2.76e6·61-s − 1.84e6·64-s + 9.63e5·71-s + 1.92e6·76-s − 2.11e6·79-s − 4.82e5·80-s − 1.12e7·89-s − 2.05e6·95-s − 7.78e6·100-s − 1.02e7·101-s − 4.02e7·109-s − 7.16e6·116-s + 1.00e8·121-s + ⋯
L(s)  = 1  + 1.09·4-s − 0.536·5-s + 3.09·11-s + 0.196·16-s + 0.458·19-s − 0.586·20-s − 0.711·25-s − 0.389·29-s + 0.990·31-s + 2.41·41-s + 3.38·44-s + 1.78·49-s − 1.66·55-s − 1.82·59-s + 1.55·61-s − 0.879·64-s + 0.319·71-s + 0.501·76-s − 0.483·79-s − 0.105·80-s − 1.69·89-s − 0.246·95-s − 0.778·100-s − 0.993·101-s − 2.97·109-s − 0.426·116-s + 5.17·121-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 197.608197.608
Root analytic conductor: 3.749313.74931
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2025, ( :7/2,7/2), 1)(4,\ 2025,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 3.9037229053.903722905
L(12)L(\frac12) ≈\approx 3.9037229053.903722905
L(92)L(\frac{9}{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
5C2C_2 1+6p2T+p7T2 1 + 6 p^{2} T + p^{7} T^{2}
good2C22C_2^2 1−35p2T2+p14T4 1 - 35 p^{2} T^{2} + p^{14} T^{4}
7C22C_2^2 1−1470650T2+p14T4 1 - 1470650 T^{2} + p^{14} T^{4}
11C2C_2 (1−6828T+p7T2)2 ( 1 - 6828 T + p^{7} T^{2} )^{2}
13C22C_2^2 1−22562810T2+p14T4 1 - 22562810 T^{2} + p^{14} T^{4}
17C22C_2^2 1−574764770T2+p14T4 1 - 574764770 T^{2} + p^{14} T^{4}
19C2C_2 (1−6860T+p7T2)2 ( 1 - 6860 T + p^{7} T^{2} )^{2}
23C22C_2^2 1−5955848090T2+p14T4 1 - 5955848090 T^{2} + p^{14} T^{4}
29C2C_2 (1+25590T+p7T2)2 ( 1 + 25590 T + p^{7} T^{2} )^{2}
31C2C_2 (1−82112T+p7T2)2 ( 1 - 82112 T + p^{7} T^{2} )^{2}
37C22C_2^2 1−139899246410T2+p14T4 1 - 139899246410 T^{2} + p^{14} T^{4}
41C2C_2 (1−533118T+p7T2)2 ( 1 - 533118 T + p^{7} T^{2} )^{2}
43C22C_2^2 1−41047812050T2+p14T4 1 - 41047812050 T^{2} + p^{14} T^{4}
47C22C_2^2 1−1013212289930T2+p14T4 1 - 1013212289930 T^{2} + p^{14} T^{4}
53C22C_2^2 1−2002060594730T2+p14T4 1 - 2002060594730 T^{2} + p^{14} T^{4}
59C2C_2 (1+1438980T+p7T2)2 ( 1 + 1438980 T + p^{7} T^{2} )^{2}
61C2C_2 (1−1381022T+p7T2)2 ( 1 - 1381022 T + p^{7} T^{2} )^{2}
67C22C_2^2 1−4750924642370T2+p14T4 1 - 4750924642370 T^{2} + p^{14} T^{4}
71C2C_2 (1−481608T+p7T2)2 ( 1 - 481608 T + p^{7} T^{2} )^{2}
73C22C_2^2 1−19886077213490T2+p14T4 1 - 19886077213490 T^{2} + p^{14} T^{4}
79C2C_2 (1+1059760T+p7T2)2 ( 1 + 1059760 T + p^{7} T^{2} )^{2}
83C22C_2^2 1−47492314121570T2+p14T4 1 - 47492314121570 T^{2} + p^{14} T^{4}
89C2C_2 (1+5644170T+p7T2)2 ( 1 + 5644170 T + p^{7} T^{2} )^{2}
97C22C_2^2 1−17378330046530T2+p14T4 1 - 17378330046530 T^{2} + p^{14} 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

−14.85850873474031755014631200360, −13.93749531021024401523168836935, −13.82952943561980649358367504562, −12.49156092167258249498568124097, −12.17186094012916599059941757749, −11.46333279955598812749814921761, −11.45677304368513362163578498724, −10.63010394160488768596802700449, −9.526725224418246340888049937428, −9.300680191322618217343810914870, −8.444145765490909196772012947001, −7.51820673238919932581996536550, −6.96331105988894652200716162107, −6.38462632311541072437273801660, −5.78175651523406429601359754259, −4.22058888768238047874571043759, −3.93174864754339985088084646245, −2.75577376786507715416427618552, −1.63558976467562824483261978922, −0.886640889462396556702910244681, 0.886640889462396556702910244681, 1.63558976467562824483261978922, 2.75577376786507715416427618552, 3.93174864754339985088084646245, 4.22058888768238047874571043759, 5.78175651523406429601359754259, 6.38462632311541072437273801660, 6.96331105988894652200716162107, 7.51820673238919932581996536550, 8.444145765490909196772012947001, 9.300680191322618217343810914870, 9.526725224418246340888049937428, 10.63010394160488768596802700449, 11.45677304368513362163578498724, 11.46333279955598812749814921761, 12.17186094012916599059941757749, 12.49156092167258249498568124097, 13.82952943561980649358367504562, 13.93749531021024401523168836935, 14.85850873474031755014631200360

Graph of the ZZ-function along the critical line