Properties

Label 2-38-19.13-c4-0-2
Degree $2$
Conductor $38$
Sign $0.956 - 0.293i$
Analytic cond. $3.92805$
Root an. cond. $1.98193$
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  + (−1.81 − 2.16i)2-s + (0.470 − 1.29i)3-s + (−1.38 + 7.87i)4-s + (8.21 + 46.5i)5-s + (−3.65 + 1.33i)6-s + (33.5 − 58.0i)7-s + (19.5 − 11.3i)8-s + (60.5 + 50.8i)9-s + (85.9 − 102. i)10-s + (3.72 + 6.44i)11-s + (9.53 + 5.50i)12-s + (77.4 + 212. i)13-s + (−186. + 32.9i)14-s + (64.0 + 11.3i)15-s + (−60.1 − 21.8i)16-s + (12.5 − 10.5i)17-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (0.0523 − 0.143i)3-s + (−0.0868 + 0.492i)4-s + (0.328 + 1.86i)5-s + (−0.101 + 0.0369i)6-s + (0.683 − 1.18i)7-s + (0.306 − 0.176i)8-s + (0.748 + 0.627i)9-s + (0.859 − 1.02i)10-s + (0.0307 + 0.0532i)11-s + (0.0662 + 0.0382i)12-s + (0.458 + 1.25i)13-s + (−0.952 + 0.167i)14-s + (0.284 + 0.0502i)15-s + (−0.234 − 0.0855i)16-s + (0.0434 − 0.0364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.293i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.956 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.956 - 0.293i$
Analytic conductor: \(3.92805\)
Root analytic conductor: \(1.98193\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ 0.956 - 0.293i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.30296 + 0.195338i\)
\(L(\frac12)\) \(\approx\) \(1.30296 + 0.195338i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.81 + 2.16i)T \)
19 \( 1 + (106. + 345. i)T \)
good3 \( 1 + (-0.470 + 1.29i)T + (-62.0 - 52.0i)T^{2} \)
5 \( 1 + (-8.21 - 46.5i)T + (-587. + 213. i)T^{2} \)
7 \( 1 + (-33.5 + 58.0i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-3.72 - 6.44i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-77.4 - 212. i)T + (-2.18e4 + 1.83e4i)T^{2} \)
17 \( 1 + (-12.5 + 10.5i)T + (1.45e4 - 8.22e4i)T^{2} \)
23 \( 1 + (130. - 740. i)T + (-2.62e5 - 9.57e4i)T^{2} \)
29 \( 1 + (-902. + 1.07e3i)T + (-1.22e5 - 6.96e5i)T^{2} \)
31 \( 1 + (649. + 374. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.15e3iT - 1.87e6T^{2} \)
41 \( 1 + (-203. + 560. i)T + (-2.16e6 - 1.81e6i)T^{2} \)
43 \( 1 + (213. + 1.20e3i)T + (-3.21e6 + 1.16e6i)T^{2} \)
47 \( 1 + (479. + 401. i)T + (8.47e5 + 4.80e6i)T^{2} \)
53 \( 1 + (-1.42e3 - 251. i)T + (7.41e6 + 2.69e6i)T^{2} \)
59 \( 1 + (2.21e3 + 2.63e3i)T + (-2.10e6 + 1.19e7i)T^{2} \)
61 \( 1 + (-214. + 1.21e3i)T + (-1.30e7 - 4.73e6i)T^{2} \)
67 \( 1 + (-2.61e3 + 3.11e3i)T + (-3.49e6 - 1.98e7i)T^{2} \)
71 \( 1 + (-4.41e3 + 777. i)T + (2.38e7 - 8.69e6i)T^{2} \)
73 \( 1 + (4.32e3 + 1.57e3i)T + (2.17e7 + 1.82e7i)T^{2} \)
79 \( 1 + (1.78e3 - 4.91e3i)T + (-2.98e7 - 2.50e7i)T^{2} \)
83 \( 1 + (3.83e3 - 6.64e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (655. + 1.80e3i)T + (-4.80e7 + 4.03e7i)T^{2} \)
97 \( 1 + (90.6 + 108. i)T + (-1.53e7 + 8.71e7i)T^{2} \)
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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

−15.57089912619482767761039484298, −14.05097246397931865129306932363, −13.59058725039296617131540120601, −11.39868279435623553866794244101, −10.78458076714723314242912550054, −9.752418536887908088718550750516, −7.62439665982097534449851762447, −6.81813305169054732729138511987, −3.99201205569928298793016299355, −2.00636115504829982315069902542, 1.25371590740096466627609651101, 4.78276613601693226259879113535, 5.90837453091628311524978542287, 8.284228926805326840238203254021, 8.809435220324017948361638339837, 10.14986180980849564641407583145, 12.20887218248136149340274299683, 12.88140071758248695060387370079, 14.63983431082347247766679738784, 15.73816741250423596017066901815

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