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Properties

Label 2-37-37.27-c7-0-3
Degree $2$
Conductor $37$
Sign $-0.777 - 0.628i$
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  + (15.8 + 9.15i)2-s + (−37.9 − 65.6i)3-s + (103. + 179. i)4-s + (−366. + 211. i)5-s − 1.38e3i·6-s + (808. + 1.40e3i)7-s + 1.45e3i·8-s + (−1.78e3 + 3.08e3i)9-s − 7.75e3·10-s − 1.95e3·11-s + (7.85e3 − 1.36e4i)12-s + (−1.05e4 + 6.11e3i)13-s + 2.96e4i·14-s + (2.77e4 + 1.60e4i)15-s + (−24.5 + 42.5i)16-s + (4.65e3 + 2.68e3i)17-s + ⋯
L(s)  = 1  + (1.40 + 0.809i)2-s + (−0.810 − 1.40i)3-s + (0.809 + 1.40i)4-s + (−1.31 + 0.757i)5-s − 2.62i·6-s + (0.890 + 1.54i)7-s + 1.00i·8-s + (−0.814 + 1.41i)9-s − 2.45·10-s − 0.443·11-s + (1.31 − 2.27i)12-s + (−1.33 + 0.772i)13-s + 2.88i·14-s + (2.12 + 1.22i)15-s + (−0.00149 + 0.00259i)16-s + (0.229 + 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.777 - 0.628i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.777 - 0.628i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.511771 + 1.44675i\)
\(L(\frac12)\) \(\approx\) \(0.511771 + 1.44675i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.43e5 + 1.89e5i)T \)
good2 \( 1 + (-15.8 - 9.15i)T + (64 + 110. i)T^{2} \)
3 \( 1 + (37.9 + 65.6i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (366. - 211. i)T + (3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-808. - 1.40e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + 1.95e3T + 1.94e7T^{2} \)
13 \( 1 + (1.05e4 - 6.11e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-4.65e3 - 2.68e3i)T + (2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (4.19e3 - 2.42e3i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + 482. iT - 3.40e9T^{2} \)
29 \( 1 + 1.68e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.38e5iT - 2.75e10T^{2} \)
41 \( 1 + (-1.20e5 - 2.08e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 1.82e5iT - 2.71e11T^{2} \)
47 \( 1 + 2.91e5T + 5.06e11T^{2} \)
53 \( 1 + (5.74e5 - 9.95e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-4.91e5 - 2.83e5i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-2.73e5 + 1.58e5i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.46e6 - 2.54e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-1.03e6 - 1.78e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 6.23e6T + 1.10e13T^{2} \)
79 \( 1 + (-2.19e6 + 1.26e6i)T + (9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-1.94e6 + 3.36e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (-4.80e6 - 2.77e6i)T + (2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 8.18e6iT - 8.07e13T^{2} \)
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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

−15.02039683555159503652369695654, −14.34758394434651388002274044056, −12.71527178656051680790472950750, −11.96215631776054804742335164797, −11.48501770488205969415940717829, −7.974898108890163668978467677175, −7.20867662677437631110408206724, −6.00926774543823860737547070229, −4.77439227142739160599477633151, −2.51255644147777895177501475532, 0.46860658978686078854786736710, 3.64114485257257604620514601268, 4.59079205061071657802597060285, 5.08933894387322520349983242397, 7.76636505540748074611444058429, 10.14460296490105036276898913001, 11.01960897463393433398848192918, 11.77158523879436920156272579797, 12.92165549909782002142336548790, 14.52620423827827000543722971315

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