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Properties

Label 2-74-37.9-c7-0-5
Degree 22
Conductor 7474
Sign −0.367−0.929i-0.367 - 0.929i
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  + (−1.38 − 7.87i)2-s + (−11.0 + 62.8i)3-s + (−60.1 + 21.8i)4-s + (48.5 + 40.7i)5-s + 510.·6-s + (745. + 625. i)7-s + (256 + 443. i)8-s + (−1.76e3 − 643. i)9-s + (253. − 439. i)10-s + (1.30e3 + 2.26e3i)11-s + (−708. − 4.02e3i)12-s + (1.08e4 − 3.93e3i)13-s + (3.89e3 − 6.74e3i)14-s + (−3.09e3 + 2.60e3i)15-s + (3.13e3 − 2.63e3i)16-s + (−1.09e4 − 3.98e3i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.236 + 1.34i)3-s + (−0.469 + 0.171i)4-s + (0.173 + 0.145i)5-s + 0.964·6-s + (0.821 + 0.689i)7-s + (0.176 + 0.306i)8-s + (−0.808 − 0.294i)9-s + (0.0802 − 0.138i)10-s + (0.296 + 0.513i)11-s + (−0.118 − 0.671i)12-s + (1.36 − 0.496i)13-s + (0.379 − 0.656i)14-s + (−0.237 + 0.198i)15-s + (0.191 − 0.160i)16-s + (−0.540 − 0.196i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.367−0.929i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.367−0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.367−0.929i-0.367 - 0.929i
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ74(9,⋅)\chi_{74} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :7/2), −0.367−0.929i)(2,\ 74,\ (\ :7/2),\ -0.367 - 0.929i)

Particular Values

L(4)L(4) ≈\approx 0.874461+1.28601i0.874461 + 1.28601i
L(12)L(\frac12) ≈\approx 0.874461+1.28601i0.874461 + 1.28601i
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+(1.38+7.87i)T 1 + (1.38 + 7.87i)T
37 1+(−2.34e5−2.00e5i)T 1 + (-2.34e5 - 2.00e5i)T
good3 1+(11.0−62.8i)T+(−2.05e3−747.i)T2 1 + (11.0 - 62.8i)T + (-2.05e3 - 747. i)T^{2}
5 1+(−48.5−40.7i)T+(1.35e4+7.69e4i)T2 1 + (-48.5 - 40.7i)T + (1.35e4 + 7.69e4i)T^{2}
7 1+(−745.−625.i)T+(1.43e5+8.11e5i)T2 1 + (-745. - 625. i)T + (1.43e5 + 8.11e5i)T^{2}
11 1+(−1.30e3−2.26e3i)T+(−9.74e6+1.68e7i)T2 1 + (-1.30e3 - 2.26e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(−1.08e4+3.93e3i)T+(4.80e7−4.03e7i)T2 1 + (-1.08e4 + 3.93e3i)T + (4.80e7 - 4.03e7i)T^{2}
17 1+(1.09e4+3.98e3i)T+(3.14e8+2.63e8i)T2 1 + (1.09e4 + 3.98e3i)T + (3.14e8 + 2.63e8i)T^{2}
19 1+(3.29e3−1.86e4i)T+(−8.39e8−3.05e8i)T2 1 + (3.29e3 - 1.86e4i)T + (-8.39e8 - 3.05e8i)T^{2}
23 1+(5.15e4−8.92e4i)T+(−1.70e9−2.94e9i)T2 1 + (5.15e4 - 8.92e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(−4.06e4−7.03e4i)T+(−8.62e9+1.49e10i)T2 1 + (-4.06e4 - 7.03e4i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+9.34e4T+2.75e10T2 1 + 9.34e4T + 2.75e10T^{2}
41 1+(−7.52e4+2.73e4i)T+(1.49e11−1.25e11i)T2 1 + (-7.52e4 + 2.73e4i)T + (1.49e11 - 1.25e11i)T^{2}
43 1+5.14e5T+2.71e11T2 1 + 5.14e5T + 2.71e11T^{2}
47 1+(4.43e5−7.67e5i)T+(−2.53e11−4.38e11i)T2 1 + (4.43e5 - 7.67e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(−1.01e6+8.54e5i)T+(2.03e11−1.15e12i)T2 1 + (-1.01e6 + 8.54e5i)T + (2.03e11 - 1.15e12i)T^{2}
59 1+(6.64e5−5.57e5i)T+(4.32e11−2.45e12i)T2 1 + (6.64e5 - 5.57e5i)T + (4.32e11 - 2.45e12i)T^{2}
61 1+(3.73e5−1.36e5i)T+(2.40e12−2.02e12i)T2 1 + (3.73e5 - 1.36e5i)T + (2.40e12 - 2.02e12i)T^{2}
67 1+(3.33e6+2.80e6i)T+(1.05e12+5.96e12i)T2 1 + (3.33e6 + 2.80e6i)T + (1.05e12 + 5.96e12i)T^{2}
71 1+(−1.29e5+7.37e5i)T+(−8.54e12−3.11e12i)T2 1 + (-1.29e5 + 7.37e5i)T + (-8.54e12 - 3.11e12i)T^{2}
73 1+3.10e6T+1.10e13T2 1 + 3.10e6T + 1.10e13T^{2}
79 1+(−3.49e6−2.92e6i)T+(3.33e12+1.89e13i)T2 1 + (-3.49e6 - 2.92e6i)T + (3.33e12 + 1.89e13i)T^{2}
83 1+(−5.92e6−2.15e6i)T+(2.07e13+1.74e13i)T2 1 + (-5.92e6 - 2.15e6i)T + (2.07e13 + 1.74e13i)T^{2}
89 1+(−3.09e6+2.59e6i)T+(7.68e12−4.35e13i)T2 1 + (-3.09e6 + 2.59e6i)T + (7.68e12 - 4.35e13i)T^{2}
97 1+(−1.06e6+1.84e6i)T+(−4.03e13−6.99e13i)T2 1 + (-1.06e6 + 1.84e6i)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

−13.44954797345657304564337973201, −11.95567438305173741076494294943, −11.12436340086055751288026138962, −10.21856498468618309101008714516, −9.243347121688003734320050832142, −8.151738883114586820081800628650, −5.86604845122200511203459637593, −4.66440888729519306243315689643, −3.51313968292395164558038701522, −1.70442996450660406232049912387, 0.60647509569657964521304559511, 1.71211312012975311157344858111, 4.24753194591520576048315692496, 5.98381443438279603664712926999, 6.82249951928643248015582590071, 7.962487331923821754367812227718, 8.862927846683014247725554007260, 10.74604322131929098938411923247, 11.74727499199929099513482938420, 13.21414585479200670763237543116

Graph of the ZZ-function along the critical line