Properties

Label 2-37-37.23-c4-0-10
Degree $2$
Conductor $37$
Sign $-0.992 + 0.120i$
Analytic cond. $3.82468$
Root an. cond. $1.95568$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.88 − 7.03i)2-s + (11.2 − 6.51i)3-s + (−32.0 + 18.4i)4-s + (9.58 − 35.7i)5-s + (−67.0 − 67.0i)6-s + (25.6 + 44.4i)7-s + (108. + 108. i)8-s + (44.3 − 76.7i)9-s − 269.·10-s + 39.1i·11-s + (−240. + 417. i)12-s + (−60.4 + 225. i)13-s + (264. − 264. i)14-s + (−124. − 466. i)15-s + (260. − 450. i)16-s + (318. − 85.3i)17-s + ⋯
L(s)  = 1  + (−0.471 − 1.75i)2-s + (1.25 − 0.723i)3-s + (−2.00 + 1.15i)4-s + (0.383 − 1.43i)5-s + (−1.86 − 1.86i)6-s + (0.524 + 0.907i)7-s + (1.68 + 1.68i)8-s + (0.547 − 0.948i)9-s − 2.69·10-s + 0.323i·11-s + (−1.67 + 2.89i)12-s + (−0.357 + 1.33i)13-s + (1.34 − 1.34i)14-s + (−0.554 − 2.07i)15-s + (1.01 − 1.76i)16-s + (1.10 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.992 + 0.120i$
Analytic conductor: \(3.82468\)
Root analytic conductor: \(1.95568\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :2),\ -0.992 + 0.120i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0961159 - 1.59221i\)
\(L(\frac12)\) \(\approx\) \(0.0961159 - 1.59221i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.27e3 - 496. i)T \)
good2 \( 1 + (1.88 + 7.03i)T + (-13.8 + 8i)T^{2} \)
3 \( 1 + (-11.2 + 6.51i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-9.58 + 35.7i)T + (-541. - 312.5i)T^{2} \)
7 \( 1 + (-25.6 - 44.4i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 - 39.1iT - 1.46e4T^{2} \)
13 \( 1 + (60.4 - 225. i)T + (-2.47e4 - 1.42e4i)T^{2} \)
17 \( 1 + (-318. + 85.3i)T + (7.23e4 - 4.17e4i)T^{2} \)
19 \( 1 + (-89.0 + 332. i)T + (-1.12e5 - 6.51e4i)T^{2} \)
23 \( 1 + (581. + 581. i)T + 2.79e5iT^{2} \)
29 \( 1 + (-109. + 109. i)T - 7.07e5iT^{2} \)
31 \( 1 + (180. - 180. i)T - 9.23e5iT^{2} \)
41 \( 1 + (1.73e3 - 1.00e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.51e3 - 1.51e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 761.T + 4.87e6T^{2} \)
53 \( 1 + (1.10e3 - 1.92e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (1.47e3 - 394. i)T + (1.04e7 - 6.05e6i)T^{2} \)
61 \( 1 + (-3.17e3 - 851. i)T + (1.19e7 + 6.92e6i)T^{2} \)
67 \( 1 + (3.77e3 - 2.17e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-4.04 - 7.00i)T + (-1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + 6.85e3iT - 2.83e7T^{2} \)
79 \( 1 + (1.43e3 - 5.36e3i)T + (-3.37e7 - 1.94e7i)T^{2} \)
83 \( 1 + (-843. + 1.46e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (-665. - 2.48e3i)T + (-5.43e7 + 3.13e7i)T^{2} \)
97 \( 1 + (751. + 751. i)T + 8.85e7iT^{2} \)
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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

−14.42639307059308220934435143332, −13.42772077922312701564554436587, −12.44667569545315843164383414591, −11.79244063203185560222222515677, −9.625050993525072407736219759895, −8.951922584915276787632285206329, −8.101526658841427496783848704935, −4.63363735285810043019566476764, −2.48074684991110360603838617251, −1.42483191043945117693933170971, 3.59629718605467793999398036927, 5.74582710723056329996878039744, 7.48669649409013186307937387762, 8.073167954189998470852171855871, 9.802156860732314059027542575328, 10.38091311544407799017518699194, 13.73801847400745803203020271720, 14.36239785673187440247306149511, 14.89328561149259057880897551873, 15.88810905019027130725753391911

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