Properties

Label 2-38-1.1-c5-0-3
Degree $2$
Conductor $38$
Sign $1$
Analytic cond. $6.09458$
Root an. cond. $2.46872$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 14.7·3-s + 16·4-s − 19.0·5-s + 59.1·6-s + 212.·7-s + 64·8-s − 23.9·9-s − 76.3·10-s − 662.·11-s + 236.·12-s + 1.11e3·13-s + 849.·14-s − 282.·15-s + 256·16-s − 522.·17-s − 95.9·18-s − 361·19-s − 305.·20-s + 3.14e3·21-s − 2.65e3·22-s − 3.21e3·23-s + 947.·24-s − 2.76e3·25-s + 4.44e3·26-s − 3.95e3·27-s + 3.39e3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.949·3-s + 0.5·4-s − 0.341·5-s + 0.671·6-s + 1.63·7-s + 0.353·8-s − 0.0987·9-s − 0.241·10-s − 1.65·11-s + 0.474·12-s + 1.82·13-s + 1.15·14-s − 0.324·15-s + 0.250·16-s − 0.438·17-s − 0.0698·18-s − 0.229·19-s − 0.170·20-s + 1.55·21-s − 1.16·22-s − 1.26·23-s + 0.335·24-s − 0.883·25-s + 1.29·26-s − 1.04·27-s + 0.818·28-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: $1$
Analytic conductor: \(6.09458\)
Root analytic conductor: \(2.46872\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.092212203\)
\(L(\frac12)\) \(\approx\) \(3.092212203\)
\(L(\frac{7}{2})\) 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 \( 1 - 14.7T + 243T^{2} \)
5 \( 1 + 19.0T + 3.12e3T^{2} \)
7 \( 1 - 212.T + 1.68e4T^{2} \)
11 \( 1 + 662.T + 1.61e5T^{2} \)
13 \( 1 - 1.11e3T + 3.71e5T^{2} \)
17 \( 1 + 522.T + 1.41e6T^{2} \)
23 \( 1 + 3.21e3T + 6.43e6T^{2} \)
29 \( 1 + 3.72e3T + 2.05e7T^{2} \)
31 \( 1 - 4.83e3T + 2.86e7T^{2} \)
37 \( 1 + 7.15e3T + 6.93e7T^{2} \)
41 \( 1 - 1.08e4T + 1.15e8T^{2} \)
43 \( 1 + 1.04e4T + 1.47e8T^{2} \)
47 \( 1 - 1.19e4T + 2.29e8T^{2} \)
53 \( 1 - 2.49e4T + 4.18e8T^{2} \)
59 \( 1 - 2.32e4T + 7.14e8T^{2} \)
61 \( 1 + 1.51e4T + 8.44e8T^{2} \)
67 \( 1 - 5.92e4T + 1.35e9T^{2} \)
71 \( 1 - 1.46e4T + 1.80e9T^{2} \)
73 \( 1 + 1.89e4T + 2.07e9T^{2} \)
79 \( 1 + 3.62e3T + 3.07e9T^{2} \)
83 \( 1 - 2.26e4T + 3.93e9T^{2} \)
89 \( 1 - 6.56e4T + 5.58e9T^{2} \)
97 \( 1 + 5.94e4T + 8.58e9T^{2} \)
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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

−15.17548649197583291338304815093, −13.99032078772933234410068781028, −13.35489327542001597598119056591, −11.61672762616855153689955877291, −10.69122895546949054067932679401, −8.421352315317977076240415353734, −7.87353626410628875190498803639, −5.57548770143407297709892494447, −3.94775800621565179734210641580, −2.15048141749809603558933299848, 2.15048141749809603558933299848, 3.94775800621565179734210641580, 5.57548770143407297709892494447, 7.87353626410628875190498803639, 8.421352315317977076240415353734, 10.69122895546949054067932679401, 11.61672762616855153689955877291, 13.35489327542001597598119056591, 13.99032078772933234410068781028, 15.17548649197583291338304815093

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