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Properties

Label 2-37-37.6-c8-0-7
Degree $2$
Conductor $37$
Sign $0.655 - 0.755i$
Analytic cond. $15.0730$
Root an. cond. $3.88239$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.65 − 2.65i)2-s − 19.9i·3-s + 241. i·4-s + (−726. − 726. i)5-s + (−52.8 − 52.8i)6-s + 711.·7-s + (1.32e3 + 1.32e3i)8-s + 6.16e3·9-s − 3.85e3·10-s + 2.08e4i·11-s + 4.81e3·12-s + (−5.82e3 − 5.82e3i)13-s + (1.88e3 − 1.88e3i)14-s + (−1.44e4 + 1.44e4i)15-s − 5.49e4·16-s + (6.82e4 + 6.82e4i)17-s + ⋯
L(s)  = 1  + (0.165 − 0.165i)2-s − 0.245i·3-s + 0.945i·4-s + (−1.16 − 1.16i)5-s + (−0.0407 − 0.0407i)6-s + 0.296·7-s + (0.322 + 0.322i)8-s + 0.939·9-s − 0.385·10-s + 1.42i·11-s + 0.232·12-s + (−0.203 − 0.203i)13-s + (0.0491 − 0.0491i)14-s + (−0.285 + 0.285i)15-s − 0.838·16-s + (0.816 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ 0.655 - 0.755i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.47388 + 0.672748i\)
\(L(\frac12)\) \(\approx\) \(1.47388 + 0.672748i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.85e6 + 2.86e5i)T \)
good2 \( 1 + (-2.65 + 2.65i)T - 256iT^{2} \)
3 \( 1 + 19.9iT - 6.56e3T^{2} \)
5 \( 1 + (726. + 726. i)T + 3.90e5iT^{2} \)
7 \( 1 - 711.T + 5.76e6T^{2} \)
11 \( 1 - 2.08e4iT - 2.14e8T^{2} \)
13 \( 1 + (5.82e3 + 5.82e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (-6.82e4 - 6.82e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (-1.32e5 - 1.32e5i)T + 1.69e10iT^{2} \)
23 \( 1 + (-1.21e5 - 1.21e5i)T + 7.83e10iT^{2} \)
29 \( 1 + (8.29e5 - 8.29e5i)T - 5.00e11iT^{2} \)
31 \( 1 + (-1.21e6 + 1.21e6i)T - 8.52e11iT^{2} \)
41 \( 1 + 1.32e6iT - 7.98e12T^{2} \)
43 \( 1 + (-1.88e6 - 1.88e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 7.67e6T + 2.38e13T^{2} \)
53 \( 1 + 3.21e6T + 6.22e13T^{2} \)
59 \( 1 + (1.61e6 + 1.61e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (3.59e6 - 3.59e6i)T - 1.91e14iT^{2} \)
67 \( 1 - 3.66e7iT - 4.06e14T^{2} \)
71 \( 1 - 7.51e6T + 6.45e14T^{2} \)
73 \( 1 + 2.18e7iT - 8.06e14T^{2} \)
79 \( 1 + (-1.95e7 - 1.95e7i)T + 1.51e15iT^{2} \)
83 \( 1 - 5.13e7T + 2.25e15T^{2} \)
89 \( 1 + (3.40e7 - 3.40e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-2.25e7 - 2.25e7i)T + 7.83e15iT^{2} \)
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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

−14.92218055540090148550075669279, −12.95839243119493301061506217794, −12.49729968671657079273502246267, −11.62174890918426800069490398497, −9.647543568965952038610534956704, −7.963441930977315125996002586713, −7.50701827047670119269885852434, −4.79924514088156873944119883805, −3.79516082324899600964932281749, −1.43666805402038314075733310550, 0.70729516510938777468135704330, 3.19352648520535877855582118604, 4.80232876175879199955397761780, 6.56203292868652575456490804787, 7.70514561400664391084173651813, 9.635980866999638952890766650314, 10.89342195097921004563750685489, 11.59220379462315609240067319788, 13.64178653691981233748686582602, 14.59647618054104277264713708738

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