Properties

Degree $2$
Conductor $38$
Sign $1$
Motivic weight $5$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 20.4·3-s + 16·4-s + 34.4·5-s − 81.9·6-s − 18.9·7-s − 64·8-s + 176.·9-s − 137.·10-s + 349.·11-s + 327.·12-s + 711.·13-s + 75.6·14-s + 705.·15-s + 256·16-s + 221.·17-s − 705.·18-s + 361·19-s + 551.·20-s − 387.·21-s − 1.39e3·22-s − 662.·23-s − 1.31e3·24-s − 1.93e3·25-s − 2.84e3·26-s − 1.36e3·27-s − 302.·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.31·3-s + 0.5·4-s + 0.616·5-s − 0.929·6-s − 0.145·7-s − 0.353·8-s + 0.726·9-s − 0.435·10-s + 0.870·11-s + 0.656·12-s + 1.16·13-s + 0.103·14-s + 0.809·15-s + 0.250·16-s + 0.186·17-s − 0.513·18-s + 0.229·19-s + 0.308·20-s − 0.191·21-s − 0.615·22-s − 0.261·23-s − 0.464·24-s − 0.620·25-s − 0.825·26-s − 0.359·27-s − 0.0729·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $1$
Motivic weight: \(5\)
Character: $\chi_{38} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.91996\)
\(L(\frac12)\) \(\approx\) \(1.91996\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
19 \( 1 - 361T \)
good3 \( 1 - 20.4T + 243T^{2} \)
5 \( 1 - 34.4T + 3.12e3T^{2} \)
7 \( 1 + 18.9T + 1.68e4T^{2} \)
11 \( 1 - 349.T + 1.61e5T^{2} \)
13 \( 1 - 711.T + 3.71e5T^{2} \)
17 \( 1 - 221.T + 1.41e6T^{2} \)
23 \( 1 + 662.T + 6.43e6T^{2} \)
29 \( 1 + 7.21e3T + 2.05e7T^{2} \)
31 \( 1 - 5.40e3T + 2.86e7T^{2} \)
37 \( 1 - 1.97e3T + 6.93e7T^{2} \)
41 \( 1 + 3.11e3T + 1.15e8T^{2} \)
43 \( 1 - 318.T + 1.47e8T^{2} \)
47 \( 1 + 2.72e4T + 2.29e8T^{2} \)
53 \( 1 + 1.11e3T + 4.18e8T^{2} \)
59 \( 1 + 3.79e4T + 7.14e8T^{2} \)
61 \( 1 - 3.74e4T + 8.44e8T^{2} \)
67 \( 1 + 5.49e4T + 1.35e9T^{2} \)
71 \( 1 + 7.17e3T + 1.80e9T^{2} \)
73 \( 1 - 6.47e4T + 2.07e9T^{2} \)
79 \( 1 - 3.61e4T + 3.07e9T^{2} \)
83 \( 1 + 5.17e4T + 3.93e9T^{2} \)
89 \( 1 - 1.45e5T + 5.58e9T^{2} \)
97 \( 1 - 3.95e4T + 8.58e9T^{2} \)
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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

−15.21649195166100810141507367349, −14.14719465065775536971790989698, −13.21687381214999115940021011408, −11.44928526567325200537447652435, −9.831642035698584421586795238501, −9.010431991225445523130268080765, −7.910684662506626978964080911307, −6.24183394782322365801246486384, −3.47819687037279433062730367506, −1.72764903657296812524078019374, 1.72764903657296812524078019374, 3.47819687037279433062730367506, 6.24183394782322365801246486384, 7.910684662506626978964080911307, 9.010431991225445523130268080765, 9.831642035698584421586795238501, 11.44928526567325200537447652435, 13.21687381214999115940021011408, 14.14719465065775536971790989698, 15.21649195166100810141507367349

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