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Properties

Label 2-74-37.7-c7-0-11
Degree 22
Conductor 7474
Sign −0.501+0.865i-0.501 + 0.865i
Analytic cond. 23.116423.1164
Root an. cond. 4.807964.80796
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.51 − 2.73i)2-s + (−83.1 − 30.2i)3-s + (49.0 − 41.1i)4-s + (77.4 + 439. i)5-s − 707.·6-s + (−71.3 − 404. i)7-s + (256. − 443. i)8-s + (4.31e3 + 3.62e3i)9-s + (1.78e3 + 3.09e3i)10-s + (3.19e3 − 5.53e3i)11-s + (−5.31e3 + 1.93e3i)12-s + (−8.36e3 + 7.01e3i)13-s + (−1.64e3 − 2.84e3i)14-s + (6.85e3 − 3.88e4i)15-s + (711. − 4.03e3i)16-s + (6.93e3 + 5.82e3i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−1.77 − 0.646i)3-s + (0.383 − 0.321i)4-s + (0.277 + 1.57i)5-s − 1.33·6-s + (−0.0786 − 0.445i)7-s + (0.176 − 0.306i)8-s + (1.97 + 1.65i)9-s + (0.564 + 0.977i)10-s + (0.723 − 1.25i)11-s + (−0.888 + 0.323i)12-s + (−1.05 + 0.885i)13-s + (−0.160 − 0.277i)14-s + (0.524 − 2.97i)15-s + (0.0434 − 0.246i)16-s + (0.342 + 0.287i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.501+0.865i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.501+0.865i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.501+0.865i-0.501 + 0.865i
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ74(7,⋅)\chi_{74} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :7/2), −0.501+0.865i)(2,\ 74,\ (\ :7/2),\ -0.501 + 0.865i)

Particular Values

L(4)L(4) ≈\approx 0.518396−0.899435i0.518396 - 0.899435i
L(12)L(\frac12) ≈\approx 0.518396−0.899435i0.518396 - 0.899435i
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}
ppFp(T)F_p(T)
bad2 1+(−7.51+2.73i)T 1 + (-7.51 + 2.73i)T
37 1+(7.04e4+2.99e5i)T 1 + (7.04e4 + 2.99e5i)T
good3 1+(83.1+30.2i)T+(1.67e3+1.40e3i)T2 1 + (83.1 + 30.2i)T + (1.67e3 + 1.40e3i)T^{2}
5 1+(−77.4−439.i)T+(−7.34e4+2.67e4i)T2 1 + (-77.4 - 439. i)T + (-7.34e4 + 2.67e4i)T^{2}
7 1+(71.3+404.i)T+(−7.73e5+2.81e5i)T2 1 + (71.3 + 404. i)T + (-7.73e5 + 2.81e5i)T^{2}
11 1+(−3.19e3+5.53e3i)T+(−9.74e6−1.68e7i)T2 1 + (-3.19e3 + 5.53e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(8.36e3−7.01e3i)T+(1.08e7−6.17e7i)T2 1 + (8.36e3 - 7.01e3i)T + (1.08e7 - 6.17e7i)T^{2}
17 1+(−6.93e3−5.82e3i)T+(7.12e7+4.04e8i)T2 1 + (-6.93e3 - 5.82e3i)T + (7.12e7 + 4.04e8i)T^{2}
19 1+(2.22e3+808.i)T+(6.84e8+5.74e8i)T2 1 + (2.22e3 + 808. i)T + (6.84e8 + 5.74e8i)T^{2}
23 1+(5.08e4+8.80e4i)T+(−1.70e9+2.94e9i)T2 1 + (5.08e4 + 8.80e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(−8.22e4+1.42e5i)T+(−8.62e9−1.49e10i)T2 1 + (-8.22e4 + 1.42e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+1.93e5T+2.75e10T2 1 + 1.93e5T + 2.75e10T^{2}
41 1+(−2.94e5+2.47e5i)T+(3.38e10−1.91e11i)T2 1 + (-2.94e5 + 2.47e5i)T + (3.38e10 - 1.91e11i)T^{2}
43 1−1.82e5T+2.71e11T2 1 - 1.82e5T + 2.71e11T^{2}
47 1+(−1.31e5−2.27e5i)T+(−2.53e11+4.38e11i)T2 1 + (-1.31e5 - 2.27e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+(−1.31e5+7.46e5i)T+(−1.10e12−4.01e11i)T2 1 + (-1.31e5 + 7.46e5i)T + (-1.10e12 - 4.01e11i)T^{2}
59 1+(1.49e4−8.45e4i)T+(−2.33e12−8.51e11i)T2 1 + (1.49e4 - 8.45e4i)T + (-2.33e12 - 8.51e11i)T^{2}
61 1+(−2.33e6+1.95e6i)T+(5.45e11−3.09e12i)T2 1 + (-2.33e6 + 1.95e6i)T + (5.45e11 - 3.09e12i)T^{2}
67 1+(6.80e4+3.85e5i)T+(−5.69e12+2.07e12i)T2 1 + (6.80e4 + 3.85e5i)T + (-5.69e12 + 2.07e12i)T^{2}
71 1+(5.49e6+1.99e6i)T+(6.96e12+5.84e12i)T2 1 + (5.49e6 + 1.99e6i)T + (6.96e12 + 5.84e12i)T^{2}
73 1+8.45e5T+1.10e13T2 1 + 8.45e5T + 1.10e13T^{2}
79 1+(−4.57e5−2.59e6i)T+(−1.80e13+6.56e12i)T2 1 + (-4.57e5 - 2.59e6i)T + (-1.80e13 + 6.56e12i)T^{2}
83 1+(8.50e5+7.13e5i)T+(4.71e12+2.67e13i)T2 1 + (8.50e5 + 7.13e5i)T + (4.71e12 + 2.67e13i)T^{2}
89 1+(−1.03e4+5.85e4i)T+(−4.15e13−1.51e13i)T2 1 + (-1.03e4 + 5.85e4i)T + (-4.15e13 - 1.51e13i)T^{2}
97 1+(8.35e6+1.44e7i)T+(−4.03e13+6.99e13i)T2 1 + (8.35e6 + 1.44e7i)T + (-4.03e13 + 6.99e13i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.50013646354547195593275706541, −11.59467839931693062812623185995, −10.88354553172813731691580756315, −10.12300832750271915555775113962, −7.29273887966035198805850891303, −6.50778782145857861843632240209, −5.81112749162626388380364456022, −4.11021693931712253239531793052, −2.16661773728995318547694333026, −0.38437910214300404961424058954, 1.29550664638881323854768007540, 4.24364636268110946984341028401, 5.14846000326333278955101971804, 5.71259540373690228682191914541, 7.26226237711689136990410851655, 9.302511856971263063757995876688, 10.15205157890496274902452260047, 11.84609665716626359419305357497, 12.23660940206451076097313777832, 12.96778995950231041213138917681

Graph of the ZZ-function along the critical line