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Properties

Label 2-37-37.27-c5-0-12
Degree $2$
Conductor $37$
Sign $0.192 + 0.981i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 + 0.570i)2-s + (2.15 + 3.73i)3-s + (−15.3 − 26.5i)4-s + (19.1 − 11.0i)5-s + 4.91i·6-s + (−51.7 − 89.6i)7-s − 71.5i·8-s + (112. − 194. i)9-s + 25.2·10-s − 61.8·11-s + (66.1 − 114. i)12-s + (628. − 363. i)13-s − 118. i·14-s + (82.4 + 47.5i)15-s + (−450. + 780. i)16-s + (−1.64e3 − 951. i)17-s + ⋯
L(s)  = 1  + (0.174 + 0.100i)2-s + (0.138 + 0.239i)3-s + (−0.479 − 0.830i)4-s + (0.342 − 0.197i)5-s + 0.0557i·6-s + (−0.399 − 0.691i)7-s − 0.395i·8-s + (0.461 − 0.799i)9-s + 0.0796·10-s − 0.154·11-s + (0.132 − 0.229i)12-s + (1.03 − 0.595i)13-s − 0.161i·14-s + (0.0946 + 0.0546i)15-s + (−0.439 + 0.761i)16-s + (−1.38 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.192 + 0.981i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.18568 - 0.975323i\)
\(L(\frac12)\) \(\approx\) \(1.18568 - 0.975323i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-8.32e3 - 294. i)T \)
good2 \( 1 + (-0.987 - 0.570i)T + (16 + 27.7i)T^{2} \)
3 \( 1 + (-2.15 - 3.73i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-19.1 + 11.0i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (51.7 + 89.6i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 61.8T + 1.61e5T^{2} \)
13 \( 1 + (-628. + 363. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (1.64e3 + 951. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.55e3 + 897. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 - 4.56e3iT - 6.43e6T^{2} \)
29 \( 1 - 282. iT - 2.05e7T^{2} \)
31 \( 1 - 237. iT - 2.86e7T^{2} \)
41 \( 1 + (-2.27e3 - 3.93e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 1.43e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.15e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.69e4 + 2.93e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-6.08e3 - 3.51e3i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.79e4 - 1.61e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.13e4 - 1.97e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.32e4 - 5.75e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 3.66e4T + 2.07e9T^{2} \)
79 \( 1 + (-996. + 575. i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.52e4 - 2.63e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-4.33e4 - 2.50e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 6.69e4iT - 8.58e9T^{2} \)
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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

−15.27352891016933334360728612815, −13.61358276460774032261877207369, −13.29338375206982565532486252228, −11.23966484919381369867498852992, −9.879002638431077810525170441342, −9.113728056992220305651598003178, −6.96112437415941349245083719653, −5.47464401117165834685637699504, −3.82648192208461812795795259641, −0.898051608951559999954593758482, 2.41882829794939014448710555430, 4.33026530511045318743252160542, 6.30068382501908119912654307452, 8.025128991684082408114867863134, 9.113869550958923390559207812000, 10.77950234011395942022878751778, 12.28423887831178252188899224851, 13.22902084592309857758377658435, 14.07689989445398478928613176766, 15.77584300509217302586029161498

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