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Properties

Label 2-37-37.26-c7-0-1
Degree $2$
Conductor $37$
Sign $0.615 - 0.788i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.57 − 13.1i)2-s + (12.8 − 22.1i)3-s + (−50.8 + 88.0i)4-s + (31.2 − 54.1i)5-s − 388.·6-s + (−403. + 698. i)7-s − 399.·8-s + (765. + 1.32e3i)9-s − 947.·10-s − 7.49e3·11-s + (1.30e3 + 2.25e3i)12-s + (−927. + 1.60e3i)13-s + 1.22e4·14-s + (−800. − 1.38e3i)15-s + (9.53e3 + 1.65e4i)16-s + (1.14e4 + 1.98e4i)17-s + ⋯
L(s)  = 1  + (−0.669 − 1.15i)2-s + (0.273 − 0.474i)3-s + (−0.397 + 0.687i)4-s + (0.111 − 0.193i)5-s − 0.733·6-s + (−0.444 + 0.769i)7-s − 0.275·8-s + (0.350 + 0.606i)9-s − 0.299·10-s − 1.69·11-s + (0.217 + 0.376i)12-s + (−0.117 + 0.202i)13-s + 1.19·14-s + (−0.0612 − 0.106i)15-s + (0.581 + 1.00i)16-s + (0.566 + 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.324928 + 0.158483i\)
\(L(\frac12)\) \(\approx\) \(0.324928 + 0.158483i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.56e5 + 2.65e5i)T \)
good2 \( 1 + (7.57 + 13.1i)T + (-64 + 110. i)T^{2} \)
3 \( 1 + (-12.8 + 22.1i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-31.2 + 54.1i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (403. - 698. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 7.49e3T + 1.94e7T^{2} \)
13 \( 1 + (927. - 1.60e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-1.14e4 - 1.98e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-2.07e4 + 3.60e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + 3.40e4T + 3.40e9T^{2} \)
29 \( 1 + 2.06e5T + 1.72e10T^{2} \)
31 \( 1 + 1.30e5T + 2.75e10T^{2} \)
41 \( 1 + (3.13e5 - 5.43e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 - 3.08e5T + 2.71e11T^{2} \)
47 \( 1 + 8.38e5T + 5.06e11T^{2} \)
53 \( 1 + (-3.20e4 - 5.54e4i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (5.32e4 + 9.21e4i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.15e5 - 2.00e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (2.33e6 - 4.04e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-3.01e5 + 5.22e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + 1.07e6T + 1.10e13T^{2} \)
79 \( 1 + (-2.10e6 + 3.65e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (2.49e6 + 4.32e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (1.71e6 + 2.96e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.31e7T + 8.07e13T^{2} \)
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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

−15.08408890295001860223723242246, −13.17281379654238546286238938287, −12.67536442635078623064374909014, −11.20043026564014893532968694142, −10.11655193034699978917839167340, −8.938440722280762129021118914619, −7.63340956641063551300202385998, −5.49651804818732933704390613014, −2.92458336311561053143565386585, −1.77287722762214942804782514284, 0.18402249487780643464263506501, 3.32020950306768127449603443403, 5.47657466536788308336857647129, 7.06204564077146371249381023507, 7.999787862886932370853521833574, 9.584024468245896266935268294237, 10.32798995828663682221349909730, 12.40014822442029498551642090219, 13.90234567604277573192404217834, 15.06447390202409641120847403663

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