Properties

Label 2-74-37.31-c4-0-11
Degree $2$
Conductor $74$
Sign $-0.726 - 0.687i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s − 5.34i·3-s + 8i·4-s + (10.1 − 10.1i)5-s + (−10.6 + 10.6i)6-s − 94.0·7-s + (16 − 16i)8-s + 52.4·9-s − 40.7·10-s − 104. i·11-s + 42.7·12-s + (−156. + 156. i)13-s + (188. + 188. i)14-s + (−54.4 − 54.4i)15-s − 64·16-s + (−288. + 288. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 0.593i·3-s + 0.5i·4-s + (0.407 − 0.407i)5-s + (−0.296 + 0.296i)6-s − 1.92·7-s + (0.250 − 0.250i)8-s + 0.647·9-s − 0.407·10-s − 0.865i·11-s + 0.296·12-s + (−0.928 + 0.928i)13-s + (0.960 + 0.960i)14-s + (−0.241 − 0.241i)15-s − 0.250·16-s + (−0.998 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ -0.726 - 0.687i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0764215 + 0.192023i\)
\(L(\frac12)\) \(\approx\) \(0.0764215 + 0.192023i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
37 \( 1 + (913. + 1.01e3i)T \)
good3 \( 1 + 5.34iT - 81T^{2} \)
5 \( 1 + (-10.1 + 10.1i)T - 625iT^{2} \)
7 \( 1 + 94.0T + 2.40e3T^{2} \)
11 \( 1 + 104. iT - 1.46e4T^{2} \)
13 \( 1 + (156. - 156. i)T - 2.85e4iT^{2} \)
17 \( 1 + (288. - 288. i)T - 8.35e4iT^{2} \)
19 \( 1 + (257. - 257. i)T - 1.30e5iT^{2} \)
23 \( 1 + (-226. + 226. i)T - 2.79e5iT^{2} \)
29 \( 1 + (59.2 + 59.2i)T + 7.07e5iT^{2} \)
31 \( 1 + (608. + 608. i)T + 9.23e5iT^{2} \)
41 \( 1 + 1.81e3iT - 2.82e6T^{2} \)
43 \( 1 + (-1.88e3 + 1.88e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 2.51e3T + 4.87e6T^{2} \)
53 \( 1 + 185.T + 7.89e6T^{2} \)
59 \( 1 + (1.98e3 - 1.98e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.80e3 + 4.80e3i)T + 1.38e7iT^{2} \)
67 \( 1 + 2.62e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.72e3T + 2.54e7T^{2} \)
73 \( 1 - 7.87e3iT - 2.83e7T^{2} \)
79 \( 1 + (-3.21e3 + 3.21e3i)T - 3.89e7iT^{2} \)
83 \( 1 + 9.87e3T + 4.74e7T^{2} \)
89 \( 1 + (-3.65e3 - 3.65e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-945. + 945. i)T - 8.85e7iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.80153914295967562438755122347, −12.46731075110566770426760302143, −10.74944302718198205043127720292, −9.646385439106593560833623169438, −8.848047337361044631904234483748, −7.12844157435768736294782743162, −6.17676702272708505280835142726, −3.84942072193650612333145556537, −2.06099650173271153187448753641, −0.11703406399883178732861041224, 2.82541742880188423561467186368, 4.77377963994803601405298060485, 6.48875639461906434805582356716, 7.21821240842939001821877723241, 9.284251286989555447111480406905, 9.802015921557124814342972162483, 10.62540770039526930089465118916, 12.57138741270628830747664221390, 13.36140408992317192074290145917, 14.93709066167101975934656503969

Graph of the $Z$-function along the critical line