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 + 265.·3-s + 256·4-s − 2.36e3·5-s − 4.24e3·6-s + 5.85e3·7-s − 4.09e3·8-s + 5.08e4·9-s + 3.78e4·10-s − 3.87e4·11-s + 6.79e4·12-s + 1.79e5·13-s − 9.37e4·14-s − 6.28e5·15-s + 6.55e4·16-s + 1.84e5·17-s − 8.13e5·18-s + 1.30e5·19-s − 6.05e5·20-s + 1.55e6·21-s + 6.19e5·22-s − 1.15e5·23-s − 1.08e6·24-s + 3.65e6·25-s − 2.87e6·26-s + 8.27e6·27-s + 1.50e6·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.89·3-s + 0.5·4-s − 1.69·5-s − 1.33·6-s + 0.922·7-s − 0.353·8-s + 2.58·9-s + 1.19·10-s − 0.797·11-s + 0.946·12-s + 1.74·13-s − 0.652·14-s − 3.20·15-s + 0.250·16-s + 0.535·17-s − 1.82·18-s + 0.229·19-s − 0.846·20-s + 1.74·21-s + 0.564·22-s − 0.0859·23-s − 0.669·24-s + 1.86·25-s − 1.23·26-s + 2.99·27-s + 0.461·28-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

\(L(5)\) \(\approx\) \(2.37156\)
\(L(\frac12)\) \(\approx\) \(2.37156\)
\(L(\frac{11}{2})\) 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 \( 1 - 265.T + 1.96e4T^{2} \)
5 \( 1 + 2.36e3T + 1.95e6T^{2} \)
7 \( 1 - 5.85e3T + 4.03e7T^{2} \)
11 \( 1 + 3.87e4T + 2.35e9T^{2} \)
13 \( 1 - 1.79e5T + 1.06e10T^{2} \)
17 \( 1 - 1.84e5T + 1.18e11T^{2} \)
23 \( 1 + 1.15e5T + 1.80e12T^{2} \)
29 \( 1 - 5.69e5T + 1.45e13T^{2} \)
31 \( 1 - 5.80e6T + 2.64e13T^{2} \)
37 \( 1 - 3.15e6T + 1.29e14T^{2} \)
41 \( 1 - 5.47e4T + 3.27e14T^{2} \)
43 \( 1 + 1.63e7T + 5.02e14T^{2} \)
47 \( 1 - 2.84e7T + 1.11e15T^{2} \)
53 \( 1 - 7.33e7T + 3.29e15T^{2} \)
59 \( 1 + 1.45e7T + 8.66e15T^{2} \)
61 \( 1 - 9.96e7T + 1.16e16T^{2} \)
67 \( 1 + 2.49e8T + 2.72e16T^{2} \)
71 \( 1 + 1.33e8T + 4.58e16T^{2} \)
73 \( 1 + 2.28e8T + 5.88e16T^{2} \)
79 \( 1 + 6.65e8T + 1.19e17T^{2} \)
83 \( 1 + 3.48e8T + 1.86e17T^{2} \)
89 \( 1 - 2.61e8T + 3.50e17T^{2} \)
97 \( 1 + 1.10e9T + 7.60e17T^{2} \)
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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

−14.58245144947934497458164348714, −13.29474783745409868351037920962, −11.72492705935312402216355867993, −10.43075849442336939118914334802, −8.643518667869504223382819281776, −8.191904380635449026958655625272, −7.39046153947799109496659904811, −4.14989855545342682112772739389, −3.00418316438181926951109022527, −1.21813914751753757329603681342, 1.21813914751753757329603681342, 3.00418316438181926951109022527, 4.14989855545342682112772739389, 7.39046153947799109496659904811, 8.191904380635449026958655625272, 8.643518667869504223382819281776, 10.43075849442336939118914334802, 11.72492705935312402216355867993, 13.29474783745409868351037920962, 14.58245144947934497458164348714

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