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Properties

Label 2-74-37.9-c7-0-16
Degree 22
Conductor 7474
Sign −0.752+0.658i-0.752 + 0.658i
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 + (−2.39 + 13.6i)3-s + (−60.1 + 21.8i)4-s + (−34.7 − 29.1i)5-s + 110.·6-s + (−140. − 117. i)7-s + (256 + 443. i)8-s + (1.87e3 + 682. i)9-s + (−181. + 313. i)10-s + (−242. − 420. i)11-s + (−153. − 871. i)12-s + (381. − 138. i)13-s + (−731. + 1.26e3i)14-s + (479. − 402. i)15-s + (3.13e3 − 2.63e3i)16-s + (−1.25e4 − 4.55e3i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.0513 + 0.291i)3-s + (−0.469 + 0.171i)4-s + (−0.124 − 0.104i)5-s + 0.208·6-s + (−0.154 − 0.129i)7-s + (0.176 + 0.306i)8-s + (0.857 + 0.312i)9-s + (−0.0573 + 0.0992i)10-s + (−0.0550 − 0.0952i)11-s + (−0.0256 − 0.145i)12-s + (0.0481 − 0.0175i)13-s + (−0.0712 + 0.123i)14-s + (0.0366 − 0.0307i)15-s + (0.191 − 0.160i)16-s + (−0.618 − 0.225i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.752+0.658i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.752+0.658i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.752+0.658i-0.752 + 0.658i
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), −0.752+0.658i)(2,\ 74,\ (\ :7/2),\ -0.752 + 0.658i)

Particular Values

L(4)L(4) ≈\approx 0.368106−0.980570i0.368106 - 0.980570i
L(12)L(\frac12) ≈\approx 0.368106−0.980570i0.368106 - 0.980570i
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+(−1.09e5+2.87e5i)T 1 + (-1.09e5 + 2.87e5i)T
good3 1+(2.39−13.6i)T+(−2.05e3−747.i)T2 1 + (2.39 - 13.6i)T + (-2.05e3 - 747. i)T^{2}
5 1+(34.7+29.1i)T+(1.35e4+7.69e4i)T2 1 + (34.7 + 29.1i)T + (1.35e4 + 7.69e4i)T^{2}
7 1+(140.+117.i)T+(1.43e5+8.11e5i)T2 1 + (140. + 117. i)T + (1.43e5 + 8.11e5i)T^{2}
11 1+(242.+420.i)T+(−9.74e6+1.68e7i)T2 1 + (242. + 420. i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(−381.+138.i)T+(4.80e7−4.03e7i)T2 1 + (-381. + 138. i)T + (4.80e7 - 4.03e7i)T^{2}
17 1+(1.25e4+4.55e3i)T+(3.14e8+2.63e8i)T2 1 + (1.25e4 + 4.55e3i)T + (3.14e8 + 2.63e8i)T^{2}
19 1+(−1.37e3+7.77e3i)T+(−8.39e8−3.05e8i)T2 1 + (-1.37e3 + 7.77e3i)T + (-8.39e8 - 3.05e8i)T^{2}
23 1+(−3.45e4+5.98e4i)T+(−1.70e9−2.94e9i)T2 1 + (-3.45e4 + 5.98e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(6.75e4+1.16e5i)T+(−8.62e9+1.49e10i)T2 1 + (6.75e4 + 1.16e5i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+1.68e5T+2.75e10T2 1 + 1.68e5T + 2.75e10T^{2}
41 1+(5.24e5−1.90e5i)T+(1.49e11−1.25e11i)T2 1 + (5.24e5 - 1.90e5i)T + (1.49e11 - 1.25e11i)T^{2}
43 1+6.32e5T+2.71e11T2 1 + 6.32e5T + 2.71e11T^{2}
47 1+(−1.61e5+2.79e5i)T+(−2.53e11−4.38e11i)T2 1 + (-1.61e5 + 2.79e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(3.91e5−3.28e5i)T+(2.03e11−1.15e12i)T2 1 + (3.91e5 - 3.28e5i)T + (2.03e11 - 1.15e12i)T^{2}
59 1+(−1.70e6+1.43e6i)T+(4.32e11−2.45e12i)T2 1 + (-1.70e6 + 1.43e6i)T + (4.32e11 - 2.45e12i)T^{2}
61 1+(−2.30e6+8.37e5i)T+(2.40e12−2.02e12i)T2 1 + (-2.30e6 + 8.37e5i)T + (2.40e12 - 2.02e12i)T^{2}
67 1+(2.51e5+2.11e5i)T+(1.05e12+5.96e12i)T2 1 + (2.51e5 + 2.11e5i)T + (1.05e12 + 5.96e12i)T^{2}
71 1+(−5.65e4+3.20e5i)T+(−8.54e12−3.11e12i)T2 1 + (-5.65e4 + 3.20e5i)T + (-8.54e12 - 3.11e12i)T^{2}
73 1−3.75e6T+1.10e13T2 1 - 3.75e6T + 1.10e13T^{2}
79 1+(1.50e6+1.26e6i)T+(3.33e12+1.89e13i)T2 1 + (1.50e6 + 1.26e6i)T + (3.33e12 + 1.89e13i)T^{2}
83 1+(−2.60e6−9.49e5i)T+(2.07e13+1.74e13i)T2 1 + (-2.60e6 - 9.49e5i)T + (2.07e13 + 1.74e13i)T^{2}
89 1+(7.70e5−6.46e5i)T+(7.68e12−4.35e13i)T2 1 + (7.70e5 - 6.46e5i)T + (7.68e12 - 4.35e13i)T^{2}
97 1+(−1.60e5+2.77e5i)T+(−4.03e13−6.99e13i)T2 1 + (-1.60e5 + 2.77e5i)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

−12.72952285834580002163348723596, −11.46247071526425522713487115158, −10.48170077712483905080184031039, −9.529844607596495701129907814752, −8.282991736847873881095269110577, −6.84029350173121728268244092385, −4.97484949696440542797581802981, −3.82594147084151037010968481648, −2.12988005010821298259916134381, −0.39208961465832784933795833784, 1.46545991108695645025032773951, 3.67816959488639912933972119993, 5.24975986405262590452420244370, 6.67251835063226838652714819105, 7.51515728537451161777018908320, 8.914879825573194583781761682403, 9.972384105906355909087997401694, 11.36552626120987448810610746394, 12.74252933019941766372271666424, 13.46761003142886480791480154949

Graph of the ZZ-function along the critical line