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Properties

Label 2-38-19.18-c6-0-2
Degree $2$
Conductor $38$
Sign $-0.862 - 0.505i$
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 + 51.5i·3-s − 32.0·4-s + 39.9·5-s + 291.·6-s − 25.1·7-s + 181. i·8-s − 1.93e3·9-s − 225. i·10-s − 1.36e3·11-s − 1.65e3i·12-s + 157. i·13-s + 142. i·14-s + 2.05e3i·15-s + 1.02e3·16-s − 2.73e3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.90i·3-s − 0.500·4-s + 0.319·5-s + 1.35·6-s − 0.0733·7-s + 0.353i·8-s − 2.64·9-s − 0.225i·10-s − 1.02·11-s − 0.954i·12-s + 0.0714i·13-s + 0.0518i·14-s + 0.609i·15-s + 0.250·16-s − 0.556·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.862 - 0.505i$
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: \((2,\ 38,\ (\ :3),\ -0.862 - 0.505i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.229241 + 0.844343i\)
\(L(\frac12)\) \(\approx\) \(0.229241 + 0.844343i\)
\(L(4)\) 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 \( 1 + (3.46e3 - 5.91e3i)T \)
good3 \( 1 - 51.5iT - 729T^{2} \)
5 \( 1 - 39.9T + 1.56e4T^{2} \)
7 \( 1 + 25.1T + 1.17e5T^{2} \)
11 \( 1 + 1.36e3T + 1.77e6T^{2} \)
13 \( 1 - 157. iT - 4.82e6T^{2} \)
17 \( 1 + 2.73e3T + 2.41e7T^{2} \)
23 \( 1 - 1.73e4T + 1.48e8T^{2} \)
29 \( 1 - 2.60e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.68e4iT - 8.87e8T^{2} \)
37 \( 1 - 8.58e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.36e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.11e5T + 6.32e9T^{2} \)
47 \( 1 - 6.06e4T + 1.07e10T^{2} \)
53 \( 1 + 1.35e3iT - 2.21e10T^{2} \)
59 \( 1 - 2.38e5iT - 4.21e10T^{2} \)
61 \( 1 - 4.29e5T + 5.15e10T^{2} \)
67 \( 1 - 1.56e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.65e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.85e5T + 1.51e11T^{2} \)
79 \( 1 + 6.06e5iT - 2.43e11T^{2} \)
83 \( 1 + 8.75e5T + 3.26e11T^{2} \)
89 \( 1 - 5.90e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.00e6iT - 8.32e11T^{2} \)
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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

−15.48312647342377039821705703203, −14.53416942818775840315467614764, −13.15561101590886258159093779692, −11.40498766811332692975388329333, −10.48864364016743508838654232152, −9.665776061951554305290697754284, −8.501315875426321651337288991251, −5.55636586905868421696696630966, −4.34254203975437924017092216848, −2.86686835526982583987332608739, 0.41397143803646312912193710737, 2.37719049816063630566633473771, 5.51385236615802927914568541628, 6.74740707413578079057923120779, 7.71290171363855685060651768427, 8.914179449848254024993546560651, 11.08383307134951674487022364139, 12.68179972974016530810242553919, 13.26604850002581477770323252137, 14.24272695902739546308209133190

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