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Properties

Label 2-74-37.7-c7-0-15
Degree 22
Conductor 7474
Sign −0.379+0.925i-0.379 + 0.925i
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 + (6.89 + 2.50i)3-s + (49.0 − 41.1i)4-s + (−27.8 − 157. i)5-s + 58.6·6-s + (117. + 664. i)7-s + (256. − 443. i)8-s + (−1.63e3 − 1.37e3i)9-s + (−641. − 1.11e3i)10-s + (2.48e3 − 4.30e3i)11-s + (441. − 160. i)12-s + (2.63e3 − 2.20e3i)13-s + (2.69e3 + 4.67e3i)14-s + (204. − 1.15e3i)15-s + (711. − 4.03e3i)16-s + (−9.59e3 − 8.05e3i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.147 + 0.0536i)3-s + (0.383 − 0.321i)4-s + (−0.0996 − 0.565i)5-s + 0.110·6-s + (0.129 + 0.732i)7-s + (0.176 − 0.306i)8-s + (−0.747 − 0.626i)9-s + (−0.202 − 0.351i)10-s + (0.563 − 0.975i)11-s + (0.0736 − 0.0268i)12-s + (0.332 − 0.278i)13-s + (0.262 + 0.455i)14-s + (0.0156 − 0.0886i)15-s + (0.0434 − 0.246i)16-s + (−0.473 − 0.397i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.379+0.925i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.379+0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.379+0.925i-0.379 + 0.925i
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), −0.379+0.925i)(2,\ 74,\ (\ :7/2),\ -0.379 + 0.925i)

Particular Values

L(4)L(4) ≈\approx 1.34351−2.00310i1.34351 - 2.00310i
L(12)L(\frac12) ≈\approx 1.34351−2.00310i1.34351 - 2.00310i
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+(1.10e5+2.87e5i)T 1 + (1.10e5 + 2.87e5i)T
good3 1+(−6.89−2.50i)T+(1.67e3+1.40e3i)T2 1 + (-6.89 - 2.50i)T + (1.67e3 + 1.40e3i)T^{2}
5 1+(27.8+157.i)T+(−7.34e4+2.67e4i)T2 1 + (27.8 + 157. i)T + (-7.34e4 + 2.67e4i)T^{2}
7 1+(−117.−664.i)T+(−7.73e5+2.81e5i)T2 1 + (-117. - 664. i)T + (-7.73e5 + 2.81e5i)T^{2}
11 1+(−2.48e3+4.30e3i)T+(−9.74e6−1.68e7i)T2 1 + (-2.48e3 + 4.30e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(−2.63e3+2.20e3i)T+(1.08e7−6.17e7i)T2 1 + (-2.63e3 + 2.20e3i)T + (1.08e7 - 6.17e7i)T^{2}
17 1+(9.59e3+8.05e3i)T+(7.12e7+4.04e8i)T2 1 + (9.59e3 + 8.05e3i)T + (7.12e7 + 4.04e8i)T^{2}
19 1+(2.38e4+8.68e3i)T+(6.84e8+5.74e8i)T2 1 + (2.38e4 + 8.68e3i)T + (6.84e8 + 5.74e8i)T^{2}
23 1+(8.80e3+1.52e4i)T+(−1.70e9+2.94e9i)T2 1 + (8.80e3 + 1.52e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(−7.13e4+1.23e5i)T+(−8.62e9−1.49e10i)T2 1 + (-7.13e4 + 1.23e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+1.66e5T+2.75e10T2 1 + 1.66e5T + 2.75e10T^{2}
41 1+(−1.50e5+1.26e5i)T+(3.38e10−1.91e11i)T2 1 + (-1.50e5 + 1.26e5i)T + (3.38e10 - 1.91e11i)T^{2}
43 1+4.51e5T+2.71e11T2 1 + 4.51e5T + 2.71e11T^{2}
47 1+(−4.26e5−7.38e5i)T+(−2.53e11+4.38e11i)T2 1 + (-4.26e5 - 7.38e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+(−1.21e5+6.91e5i)T+(−1.10e12−4.01e11i)T2 1 + (-1.21e5 + 6.91e5i)T + (-1.10e12 - 4.01e11i)T^{2}
59 1+(2.22e5−1.26e6i)T+(−2.33e12−8.51e11i)T2 1 + (2.22e5 - 1.26e6i)T + (-2.33e12 - 8.51e11i)T^{2}
61 1+(−2.24e5+1.88e5i)T+(5.45e11−3.09e12i)T2 1 + (-2.24e5 + 1.88e5i)T + (5.45e11 - 3.09e12i)T^{2}
67 1+(−4.77e5−2.70e6i)T+(−5.69e12+2.07e12i)T2 1 + (-4.77e5 - 2.70e6i)T + (-5.69e12 + 2.07e12i)T^{2}
71 1+(−3.46e6−1.26e6i)T+(6.96e12+5.84e12i)T2 1 + (-3.46e6 - 1.26e6i)T + (6.96e12 + 5.84e12i)T^{2}
73 1−1.21e6T+1.10e13T2 1 - 1.21e6T + 1.10e13T^{2}
79 1+(−7.62e4−4.32e5i)T+(−1.80e13+6.56e12i)T2 1 + (-7.62e4 - 4.32e5i)T + (-1.80e13 + 6.56e12i)T^{2}
83 1+(−3.94e6−3.31e6i)T+(4.71e12+2.67e13i)T2 1 + (-3.94e6 - 3.31e6i)T + (4.71e12 + 2.67e13i)T^{2}
89 1+(1.34e6−7.65e6i)T+(−4.15e13−1.51e13i)T2 1 + (1.34e6 - 7.65e6i)T + (-4.15e13 - 1.51e13i)T^{2}
97 1+(−7.21e6−1.25e7i)T+(−4.03e13+6.99e13i)T2 1 + (-7.21e6 - 1.25e7i)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

−12.71876766030534410157126308855, −11.81126399320905457070617556268, −10.91291711934051429337880006317, −9.146744823446519417736382426760, −8.449648745826992237243494014577, −6.44416289003070100781924541269, −5.43805736327235230171212415523, −3.92436303051668831131026815176, −2.53254938651782243113036113934, −0.63405505879536210548699421947, 1.91793013319248001785317279445, 3.53795749225543763849379988144, 4.81016053506904420655879523538, 6.44319598955102755171566503935, 7.38722535297602775067033604618, 8.704628305704977028349254570893, 10.43094298697956601769653580560, 11.26207212628816132281294147140, 12.53153417113118506185604163948, 13.67444236162597477469849856355

Graph of the ZZ-function along the critical line