Properties

Degree $2$
Conductor $38$
Sign $-1$
Motivic weight $7$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 72.2·3-s + 64·4-s + 467.·5-s − 577.·6-s − 1.47e3·7-s + 512·8-s + 3.03e3·9-s + 3.73e3·10-s − 3.54e3·11-s − 4.62e3·12-s − 1.23e4·13-s − 1.17e4·14-s − 3.37e4·15-s + 4.09e3·16-s − 1.34e4·17-s + 2.42e4·18-s − 6.85e3·19-s + 2.99e4·20-s + 1.06e5·21-s − 2.83e4·22-s − 9.40e4·23-s − 3.69e4·24-s + 1.40e5·25-s − 9.88e4·26-s − 6.10e4·27-s − 9.41e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.54·3-s + 0.5·4-s + 1.67·5-s − 1.09·6-s − 1.62·7-s + 0.353·8-s + 1.38·9-s + 1.18·10-s − 0.803·11-s − 0.772·12-s − 1.56·13-s − 1.14·14-s − 2.58·15-s + 0.250·16-s − 0.665·17-s + 0.980·18-s − 0.229·19-s + 0.836·20-s + 2.50·21-s − 0.568·22-s − 1.61·23-s − 0.546·24-s + 1.79·25-s − 1.10·26-s − 0.596·27-s − 0.810·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-1$
Motivic weight: \(7\)
Character: $\chi_{38} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8T \)
19 \( 1 + 6.85e3T \)
good3 \( 1 + 72.2T + 2.18e3T^{2} \)
5 \( 1 - 467.T + 7.81e4T^{2} \)
7 \( 1 + 1.47e3T + 8.23e5T^{2} \)
11 \( 1 + 3.54e3T + 1.94e7T^{2} \)
13 \( 1 + 1.23e4T + 6.27e7T^{2} \)
17 \( 1 + 1.34e4T + 4.10e8T^{2} \)
23 \( 1 + 9.40e4T + 3.40e9T^{2} \)
29 \( 1 - 3.35e4T + 1.72e10T^{2} \)
31 \( 1 - 2.12e5T + 2.75e10T^{2} \)
37 \( 1 - 3.36e4T + 9.49e10T^{2} \)
41 \( 1 + 4.80e5T + 1.94e11T^{2} \)
43 \( 1 + 5.61e5T + 2.71e11T^{2} \)
47 \( 1 - 4.78e5T + 5.06e11T^{2} \)
53 \( 1 + 9.31e4T + 1.17e12T^{2} \)
59 \( 1 - 9.55e5T + 2.48e12T^{2} \)
61 \( 1 + 1.71e6T + 3.14e12T^{2} \)
67 \( 1 - 4.76e5T + 6.06e12T^{2} \)
71 \( 1 - 1.95e6T + 9.09e12T^{2} \)
73 \( 1 - 1.21e6T + 1.10e13T^{2} \)
79 \( 1 - 3.94e6T + 1.92e13T^{2} \)
83 \( 1 + 3.54e6T + 2.71e13T^{2} \)
89 \( 1 + 7.50e6T + 4.42e13T^{2} \)
97 \( 1 - 1.27e7T + 8.07e13T^{2} \)
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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

−13.79386770200144625299144922712, −12.89822855690209524814218253750, −12.08202323532062773319172066580, −10.28772306409231108133482774754, −9.886381209645362185797124064715, −6.71448189102433090559053016916, −6.02667985244290819787978193283, −4.97474178816789664448980853352, −2.42938874675819712221643987754, 0, 2.42938874675819712221643987754, 4.97474178816789664448980853352, 6.02667985244290819787978193283, 6.71448189102433090559053016916, 9.886381209645362185797124064715, 10.28772306409231108133482774754, 12.08202323532062773319172066580, 12.89822855690209524814218253750, 13.79386770200144625299144922712

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