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Properties

Label 2-38-19.7-c3-0-3
Degree $2$
Conductor $38$
Sign $-0.0336 + 0.999i$
Analytic cond. $2.24207$
Root an. cond. $1.49735$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−4.78 − 8.29i)3-s + (−1.99 + 3.46i)4-s + (−7.88 − 13.6i)5-s + (9.57 − 16.5i)6-s + 16.5·7-s − 7.99·8-s + (−32.3 + 56.0i)9-s + (15.7 − 27.3i)10-s + 16.0·11-s + 38.3·12-s + (33.5 − 58.1i)13-s + (16.5 + 28.7i)14-s + (−75.5 + 130. i)15-s + (−8 − 13.8i)16-s + (−19.8 − 34.2i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.921 − 1.59i)3-s + (−0.249 + 0.433i)4-s + (−0.705 − 1.22i)5-s + (0.651 − 1.12i)6-s + 0.895·7-s − 0.353·8-s + (−1.19 + 2.07i)9-s + (0.498 − 0.864i)10-s + 0.439·11-s + 0.921·12-s + (0.715 − 1.23i)13-s + (0.316 + 0.548i)14-s + (−1.30 + 2.25i)15-s + (−0.125 − 0.216i)16-s + (−0.282 − 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0336 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0336 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.0336 + 0.999i$
Analytic conductor: \(2.24207\)
Root analytic conductor: \(1.49735\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ -0.0336 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.698241 - 0.722126i\)
\(L(\frac12)\) \(\approx\) \(0.698241 - 0.722126i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
19 \( 1 + (-14.7 - 81.4i)T \)
good3 \( 1 + (4.78 + 8.29i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (7.88 + 13.6i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 16.5T + 343T^{2} \)
11 \( 1 - 16.0T + 1.33e3T^{2} \)
13 \( 1 + (-33.5 + 58.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (19.8 + 34.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-23.8 + 41.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-59.2 + 102. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 120.T + 2.97e4T^{2} \)
37 \( 1 - 22.0T + 5.06e4T^{2} \)
41 \( 1 + (54.5 + 94.4i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-180. - 312. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (96.4 - 167. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (108. - 188. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (211. + 366. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (281. - 488. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-471. + 816. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-332. - 576. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-93.4 - 161. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-438. - 758. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 476.T + 5.71e5T^{2} \)
89 \( 1 + (-482. + 835. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-792. - 1.37e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.75741060421122863395275973578, −14.06152623544195647766069222205, −12.92369056749836021922546449345, −12.21372071895760156674016176291, −11.26099312374945797982516773625, −8.333555476063524327034207357977, −7.75368752250824259559537062322, −6.10127702843787077427740460989, −4.86246694116509763839456969682, −0.951944896373544742709307279213, 3.63241632286615355302515022522, 4.73547993877196561568401551343, 6.51743092315282708339918372977, 9.019354343058023067979772174920, 10.48807147033820685828028596373, 11.27919357588992321344695556608, 11.69521901217078936702110278071, 14.14267436357672101147879284635, 15.02368399167569656558165558351, 15.82715454977026008750706071501

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