Properties

Label 2-38-19.18-c6-0-3
Degree $2$
Conductor $38$
Sign $0.989 + 0.144i$
Analytic cond. $8.74205$
Root an. cond. $2.95669$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65i·2-s + 14.3i·3-s − 32.0·4-s − 88.1·5-s + 81.2·6-s + 443.·7-s + 181. i·8-s + 522.·9-s + 498. i·10-s + 843.·11-s − 459. i·12-s + 2.24e3i·13-s − 2.50e3i·14-s − 1.26e3i·15-s + 1.02e3·16-s + 8.66e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.532i·3-s − 0.500·4-s − 0.705·5-s + 0.376·6-s + 1.29·7-s + 0.353i·8-s + 0.716·9-s + 0.498i·10-s + 0.633·11-s − 0.266i·12-s + 1.02i·13-s − 0.913i·14-s − 0.375i·15-s + 0.250·16-s + 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(8.74205\)
Root analytic conductor: \(2.95669\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.74626 - 0.127159i\)
\(L(\frac12)\) \(\approx\) \(1.74626 - 0.127159i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65iT \)
19 \( 1 + (-993. + 6.78e3i)T \)
good3 \( 1 - 14.3iT - 729T^{2} \)
5 \( 1 + 88.1T + 1.56e4T^{2} \)
7 \( 1 - 443.T + 1.17e5T^{2} \)
11 \( 1 - 843.T + 1.77e6T^{2} \)
13 \( 1 - 2.24e3iT - 4.82e6T^{2} \)
17 \( 1 - 8.66e3T + 2.41e7T^{2} \)
23 \( 1 + 4.04e3T + 1.48e8T^{2} \)
29 \( 1 - 1.20e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.29e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.14e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.34e5iT - 4.75e9T^{2} \)
43 \( 1 - 2.84e4T + 6.32e9T^{2} \)
47 \( 1 - 4.08e4T + 1.07e10T^{2} \)
53 \( 1 + 1.53e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.87e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.62e5T + 5.15e10T^{2} \)
67 \( 1 + 5.10e4iT - 9.04e10T^{2} \)
71 \( 1 + 4.85e5iT - 1.28e11T^{2} \)
73 \( 1 + 7.05e5T + 1.51e11T^{2} \)
79 \( 1 + 5.79e4iT - 2.43e11T^{2} \)
83 \( 1 + 1.74e5T + 3.26e11T^{2} \)
89 \( 1 + 4.46e5iT - 4.96e11T^{2} \)
97 \( 1 + 5.03e5iT - 8.32e11T^{2} \)
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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

−14.86255612662393206567381448384, −13.94556064812438904526002107626, −12.15448458825282269555987954404, −11.45348297748849687821185944114, −10.19711817670030501964944273947, −8.853484412730312183070889393435, −7.42106034765480108162797407306, −4.91331465708428092928447319842, −3.80495065365644384940558674838, −1.42248295170733489409069884644, 1.16626919971685974766930555358, 4.06428650327858334560016813089, 5.73528580192971317406601437625, 7.64794114181523467393270731369, 7.983311737315568529823260955623, 9.980592874675244892645845007735, 11.65773543159268229180311709822, 12.66033440895313232367002935533, 14.17881274993847167081262883598, 14.96881238104270612268969927479

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