Properties

Label 2-37-37.10-c7-0-5
Degree $2$
Conductor $37$
Sign $0.736 - 0.676i$
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  + (−5.02 + 8.70i)2-s + (−17.3 − 30.0i)3-s + (13.4 + 23.3i)4-s + (14.1 + 24.5i)5-s + 348.·6-s + (−329. − 571. i)7-s − 1.55e3·8-s + (492. − 853. i)9-s − 285.·10-s + 6.67e3·11-s + (466. − 807. i)12-s + (3.45e3 + 5.98e3i)13-s + 6.63e3·14-s + (491. − 851. i)15-s + (6.10e3 − 1.05e4i)16-s + (−7.84e3 + 1.35e4i)17-s + ⋯
L(s)  = 1  + (−0.444 + 0.769i)2-s + (−0.370 − 0.641i)3-s + (0.105 + 0.182i)4-s + (0.0507 + 0.0878i)5-s + 0.658·6-s + (−0.363 − 0.629i)7-s − 1.07·8-s + (0.225 − 0.390i)9-s − 0.0901·10-s + 1.51·11-s + (0.0779 − 0.134i)12-s + (0.435 + 0.755i)13-s + 0.646·14-s + (0.0376 − 0.0651i)15-s + (0.372 − 0.645i)16-s + (−0.387 + 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.736 - 0.676i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.736 - 0.676i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.25428 + 0.488476i\)
\(L(\frac12)\) \(\approx\) \(1.25428 + 0.488476i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.77e5 - 1.34e5i)T \)
good2 \( 1 + (5.02 - 8.70i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (17.3 + 30.0i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-14.1 - 24.5i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (329. + 571. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 6.67e3T + 1.94e7T^{2} \)
13 \( 1 + (-3.45e3 - 5.98e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (7.84e3 - 1.35e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-2.32e4 - 4.02e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 6.93e4T + 3.40e9T^{2} \)
29 \( 1 - 9.17e4T + 1.72e10T^{2} \)
31 \( 1 - 2.24e5T + 2.75e10T^{2} \)
41 \( 1 + (1.78e5 + 3.08e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + 9.10e5T + 2.71e11T^{2} \)
47 \( 1 + 8.53e4T + 5.06e11T^{2} \)
53 \( 1 + (2.15e5 - 3.72e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-6.74e5 + 1.16e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (3.41e5 + 5.91e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.45e6 - 2.51e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (1.39e6 + 2.42e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 6.51e6T + 1.10e13T^{2} \)
79 \( 1 + (-2.48e6 - 4.30e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (3.11e6 - 5.40e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (8.09e5 - 1.40e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 1.78e5T + 8.07e13T^{2} \)
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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

−15.15766668871333366481172460598, −13.91359468965024969244717795374, −12.43996701540972111240650296733, −11.59664670062854996233924597516, −9.711196198287921811340214733409, −8.375615555110678950866062493757, −6.75941285135080160797748812050, −6.46455929440540191784542338427, −3.74027830421757961840042148119, −1.10350780849201996810378198650, 1.00314315102779774818700958971, 2.97197017861083339195013086412, 5.04240528886671212675697028574, 6.59167799552746026815672462518, 8.961585010869585719522049103971, 9.741445155647025969109511768085, 11.05645644713847485903709150324, 11.77718609509650929550248149153, 13.31067063054061762937222019370, 15.03612478256878156195138345933

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