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Properties

Label 2-37-37.31-c8-0-3
Degree $2$
Conductor $37$
Sign $-0.703 + 0.710i$
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  + (11.9 + 11.9i)2-s + 129. i·3-s + 28.7i·4-s + (−176. + 176. i)5-s + (−1.55e3 + 1.55e3i)6-s − 2.76e3·7-s + (2.71e3 − 2.71e3i)8-s − 1.03e4·9-s − 4.21e3·10-s + 5.59e3i·11-s − 3.73e3·12-s + (−2.07e4 + 2.07e4i)13-s + (−3.29e4 − 3.29e4i)14-s + (−2.29e4 − 2.29e4i)15-s + 7.20e4·16-s + (7.88e4 − 7.88e4i)17-s + ⋯
L(s)  = 1  + (0.745 + 0.745i)2-s + 1.60i·3-s + 0.112i·4-s + (−0.282 + 0.282i)5-s + (−1.19 + 1.19i)6-s − 1.15·7-s + (0.662 − 0.662i)8-s − 1.57·9-s − 0.421·10-s + 0.382i·11-s − 0.179·12-s + (−0.726 + 0.726i)13-s + (−0.857 − 0.857i)14-s + (−0.453 − 0.453i)15-s + 1.09·16-s + (0.944 − 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.703 + 0.710i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ -0.703 + 0.710i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.539711 - 1.29341i\)
\(L(\frac12)\) \(\approx\) \(0.539711 - 1.29341i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.44e5 - 1.86e6i)T \)
good2 \( 1 + (-11.9 - 11.9i)T + 256iT^{2} \)
3 \( 1 - 129. iT - 6.56e3T^{2} \)
5 \( 1 + (176. - 176. i)T - 3.90e5iT^{2} \)
7 \( 1 + 2.76e3T + 5.76e6T^{2} \)
11 \( 1 - 5.59e3iT - 2.14e8T^{2} \)
13 \( 1 + (2.07e4 - 2.07e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-7.88e4 + 7.88e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (3.12e4 - 3.12e4i)T - 1.69e10iT^{2} \)
23 \( 1 + (1.72e5 - 1.72e5i)T - 7.83e10iT^{2} \)
29 \( 1 + (5.76e5 + 5.76e5i)T + 5.00e11iT^{2} \)
31 \( 1 + (3.06e5 + 3.06e5i)T + 8.52e11iT^{2} \)
41 \( 1 - 4.57e6iT - 7.98e12T^{2} \)
43 \( 1 + (1.73e6 - 1.73e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 1.08e6T + 2.38e13T^{2} \)
53 \( 1 - 1.34e7T + 6.22e13T^{2} \)
59 \( 1 + (4.21e6 - 4.21e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (-9.22e6 - 9.22e6i)T + 1.91e14iT^{2} \)
67 \( 1 - 1.69e7iT - 4.06e14T^{2} \)
71 \( 1 + 2.90e7T + 6.45e14T^{2} \)
73 \( 1 + 2.25e7iT - 8.06e14T^{2} \)
79 \( 1 + (1.42e7 - 1.42e7i)T - 1.51e15iT^{2} \)
83 \( 1 - 4.36e7T + 2.25e15T^{2} \)
89 \( 1 + (5.77e6 + 5.77e6i)T + 3.93e15iT^{2} \)
97 \( 1 + (-5.99e7 + 5.99e7i)T - 7.83e15iT^{2} \)
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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

−15.23367963463725845608082199090, −14.66468610545235279960371872365, −13.36150088690890544217904987803, −11.67236934166914335816114255541, −9.998696933263428938400559438752, −9.599901834268091913463035407849, −7.26045405613011844742000796053, −5.76899264657742778502544173613, −4.51170406499446126416413831605, −3.38281668019045622093931644794, 0.42350481733171698343555285128, 2.17489255036178466981225014843, 3.51943437119156422895852510498, 5.72497662342093406406132633328, 7.23350958749162094602689199976, 8.390649229899411533355293744270, 10.51735981203790414690982544140, 12.23783473250704712602569564629, 12.47456258308123822276356317469, 13.33944562036290486697815132485

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