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Properties

Label 2-37-37.36-c7-0-7
Degree $2$
Conductor $37$
Sign $0.963 - 0.268i$
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  + 18.8i·2-s − 54.3·3-s − 228.·4-s − 103. i·5-s − 1.02e3i·6-s − 202.·7-s − 1.89e3i·8-s + 772.·9-s + 1.95e3·10-s + 2.66e3·11-s + 1.24e4·12-s − 336. i·13-s − 3.82e3i·14-s + 5.63e3i·15-s + 6.57e3·16-s − 5.88e3i·17-s + ⋯
L(s)  = 1  + 1.66i·2-s − 1.16·3-s − 1.78·4-s − 0.370i·5-s − 1.94i·6-s − 0.223·7-s − 1.30i·8-s + 0.353·9-s + 0.618·10-s + 0.602·11-s + 2.07·12-s − 0.0424i·13-s − 0.372i·14-s + 0.430i·15-s + 0.401·16-s − 0.290i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.963 - 0.268i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.963 - 0.268i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.605221 + 0.0827767i\)
\(L(\frac12)\) \(\approx\) \(0.605221 + 0.0827767i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.96e5 + 8.27e4i)T \)
good2 \( 1 - 18.8iT - 128T^{2} \)
3 \( 1 + 54.3T + 2.18e3T^{2} \)
5 \( 1 + 103. iT - 7.81e4T^{2} \)
7 \( 1 + 202.T + 8.23e5T^{2} \)
11 \( 1 - 2.66e3T + 1.94e7T^{2} \)
13 \( 1 + 336. iT - 6.27e7T^{2} \)
17 \( 1 + 5.88e3iT - 4.10e8T^{2} \)
19 \( 1 - 1.63e4iT - 8.93e8T^{2} \)
23 \( 1 + 2.68e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.68e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.97e5iT - 2.75e10T^{2} \)
41 \( 1 + 2.52e5T + 1.94e11T^{2} \)
43 \( 1 + 2.02e5iT - 2.71e11T^{2} \)
47 \( 1 - 3.28e4T + 5.06e11T^{2} \)
53 \( 1 + 1.37e6T + 1.17e12T^{2} \)
59 \( 1 - 2.89e4iT - 2.48e12T^{2} \)
61 \( 1 + 2.40e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.27e6T + 6.06e12T^{2} \)
71 \( 1 - 2.07e6T + 9.09e12T^{2} \)
73 \( 1 + 1.38e6T + 1.10e13T^{2} \)
79 \( 1 - 4.15e6iT - 1.92e13T^{2} \)
83 \( 1 - 3.19e6T + 2.71e13T^{2} \)
89 \( 1 - 8.56e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.46e7iT - 8.07e13T^{2} \)
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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

−15.19570012459004417228634719994, −14.03746754018559911528465024690, −12.66447554231510539389041132957, −11.33737301156204998990390050388, −9.555402577769884323380152124885, −8.143354664147934369323461371652, −6.65233558987585314855220099421, −5.79148536833348032373366880461, −4.55406718452798871546749004615, −0.37293719705110555720332956572, 1.19851139802019810175509621863, 3.15973766580433334089177874555, 4.85513326890721121807439162605, 6.59953785145121422426307552328, 9.031980147826384936607977139723, 10.39111863151358264637825957704, 11.17232289330389865498322651793, 12.04045332016627040415167847745, 13.01385229792324030716087989660, 14.44967041560743066172367086810

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