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Properties

Label 2-37-37.27-c5-0-15
Degree $2$
Conductor $37$
Sign $-0.869 - 0.494i$
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  + (−3.47 − 2.00i)2-s + (−13.3 − 23.0i)3-s + (−7.96 − 13.7i)4-s + (85.3 − 49.2i)5-s + 106. i·6-s + (−91.8 − 159. i)7-s + 192. i·8-s + (−233. + 404. i)9-s − 395.·10-s + 64.5·11-s + (−212. + 367. i)12-s + (582. − 336. i)13-s + 736. i·14-s + (−2.27e3 − 1.31e3i)15-s + (130. − 225. i)16-s + (1.57e3 + 911. i)17-s + ⋯
L(s)  = 1  + (−0.613 − 0.354i)2-s + (−0.854 − 1.47i)3-s + (−0.248 − 0.431i)4-s + (1.52 − 0.881i)5-s + 1.21i·6-s + (−0.708 − 1.22i)7-s + 1.06i·8-s + (−0.960 + 1.66i)9-s − 1.24·10-s + 0.160·11-s + (−0.425 + 0.736i)12-s + (0.955 − 0.551i)13-s + 1.00i·14-s + (−2.60 − 1.50i)15-s + (0.127 − 0.220i)16-s + (1.32 + 0.764i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.869 - 0.494i$
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: \((2,\ 37,\ (\ :5/2),\ -0.869 - 0.494i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.224039 + 0.847076i\)
\(L(\frac12)\) \(\approx\) \(0.224039 + 0.847076i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (5.65e3 - 6.11e3i)T \)
good2 \( 1 + (3.47 + 2.00i)T + (16 + 27.7i)T^{2} \)
3 \( 1 + (13.3 + 23.0i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-85.3 + 49.2i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (91.8 + 159. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 64.5T + 1.61e5T^{2} \)
13 \( 1 + (-582. + 336. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-1.57e3 - 911. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (262. - 151. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 378. iT - 6.43e6T^{2} \)
29 \( 1 + 2.89e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.34e3iT - 2.86e7T^{2} \)
41 \( 1 + (1.24e3 + 2.15e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 3.20e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.42e3T + 2.29e8T^{2} \)
53 \( 1 + (-6.15e3 + 1.06e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.56e4 + 1.48e4i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.62e4 - 1.51e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-7.24e3 - 1.25e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-2.61e4 - 4.52e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 6.89e4T + 2.07e9T^{2} \)
79 \( 1 + (-7.36e4 + 4.25e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (3.48e4 - 6.03e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (6.26e4 + 3.61e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 6.81e4iT - 8.58e9T^{2} \)
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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

−13.95297977781207844619322128298, −13.38516236181903127197611248162, −12.46348345496092001194858763129, −10.71348582023497607462646572276, −9.843628236806809154870092986663, −8.229662802197936175237547456363, −6.39310248890101138098784881348, −5.51923509125579851376514957434, −1.52364292165881886744224676235, −0.76551870751378767348230683123, 3.29672181332474200291355111062, 5.51433706898679507402009314235, 6.47929404054759805242183585518, 9.174497508784601401382231276428, 9.568936580165283517964021099373, 10.70790015461661882723171902448, 12.23325413768219435323112014759, 13.89464529912441854855167986045, 15.29443889844243244084829332060, 16.28404958341713754801752282635

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