Properties

Degree $2$
Conductor $38$
Sign $1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 25.2·3-s + 256·4-s + 2.12e3·5-s − 404.·6-s + 1.14e4·7-s − 4.09e3·8-s − 1.90e4·9-s − 3.40e4·10-s − 1.04e4·11-s + 6.46e3·12-s + 1.44e5·13-s − 1.83e5·14-s + 5.37e4·15-s + 6.55e4·16-s − 6.54e5·17-s + 3.04e5·18-s + 1.30e5·19-s + 5.44e5·20-s + 2.89e5·21-s + 1.66e5·22-s + 1.63e6·23-s − 1.03e5·24-s + 2.56e6·25-s − 2.31e6·26-s − 9.78e5·27-s + 2.93e6·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.180·3-s + 0.5·4-s + 1.52·5-s − 0.127·6-s + 1.80·7-s − 0.353·8-s − 0.967·9-s − 1.07·10-s − 0.214·11-s + 0.0900·12-s + 1.40·13-s − 1.27·14-s + 0.273·15-s + 0.250·16-s − 1.89·17-s + 0.684·18-s + 0.229·19-s + 0.760·20-s + 0.325·21-s + 0.151·22-s + 1.21·23-s − 0.0636·24-s + 1.31·25-s − 0.993·26-s − 0.354·27-s + 0.902·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $1$
Motivic weight: \(9\)
Character: $\chi_{38} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(2.21899\)
\(L(\frac12)\) \(\approx\) \(2.21899\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16T \)
19 \( 1 - 1.30e5T \)
good3 \( 1 - 25.2T + 1.96e4T^{2} \)
5 \( 1 - 2.12e3T + 1.95e6T^{2} \)
7 \( 1 - 1.14e4T + 4.03e7T^{2} \)
11 \( 1 + 1.04e4T + 2.35e9T^{2} \)
13 \( 1 - 1.44e5T + 1.06e10T^{2} \)
17 \( 1 + 6.54e5T + 1.18e11T^{2} \)
23 \( 1 - 1.63e6T + 1.80e12T^{2} \)
29 \( 1 - 3.60e6T + 1.45e13T^{2} \)
31 \( 1 - 3.62e6T + 2.64e13T^{2} \)
37 \( 1 + 1.06e7T + 1.29e14T^{2} \)
41 \( 1 - 1.31e7T + 3.27e14T^{2} \)
43 \( 1 + 2.73e6T + 5.02e14T^{2} \)
47 \( 1 + 5.26e7T + 1.11e15T^{2} \)
53 \( 1 - 5.87e7T + 3.29e15T^{2} \)
59 \( 1 + 7.23e7T + 8.66e15T^{2} \)
61 \( 1 + 7.46e7T + 1.16e16T^{2} \)
67 \( 1 - 1.12e8T + 2.72e16T^{2} \)
71 \( 1 - 3.11e8T + 4.58e16T^{2} \)
73 \( 1 - 3.47e6T + 5.88e16T^{2} \)
79 \( 1 - 1.96e8T + 1.19e17T^{2} \)
83 \( 1 + 3.30e8T + 1.86e17T^{2} \)
89 \( 1 + 6.40e8T + 3.50e17T^{2} \)
97 \( 1 - 1.56e9T + 7.60e17T^{2} \)
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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

−14.19369639544271822047502192949, −13.43687981441028757259676519291, −11.35526544113660772909640348410, −10.70979808243567990751302295199, −8.997917585349156301939243383511, −8.356835922583964724022482897467, −6.42444062838758251134464802650, −5.09102612393007104888397543886, −2.40668740999092649516449211382, −1.30234077860475523544203894450, 1.30234077860475523544203894450, 2.40668740999092649516449211382, 5.09102612393007104888397543886, 6.42444062838758251134464802650, 8.356835922583964724022482897467, 8.997917585349156301939243383511, 10.70979808243567990751302295199, 11.35526544113660772909640348410, 13.43687981441028757259676519291, 14.19369639544271822047502192949

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