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Properties

Label 2-37-37.26-c5-0-13
Degree $2$
Conductor $37$
Sign $-0.388 + 0.921i$
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  + (−0.480 − 0.832i)2-s + (12.5 − 21.7i)3-s + (15.5 − 26.9i)4-s + (14.2 − 24.6i)5-s − 24.1·6-s + (−96.5 + 167. i)7-s − 60.6·8-s + (−193. − 335. i)9-s − 27.3·10-s + 478.·11-s + (−390. − 675. i)12-s + (−111. + 192. i)13-s + 185.·14-s + (−357. − 619. i)15-s + (−468. − 810. i)16-s + (447. + 775. i)17-s + ⋯
L(s)  = 1  + (−0.0849 − 0.147i)2-s + (0.805 − 1.39i)3-s + (0.485 − 0.841i)4-s + (0.254 − 0.441i)5-s − 0.273·6-s + (−0.744 + 1.29i)7-s − 0.334·8-s + (−0.797 − 1.38i)9-s − 0.0866·10-s + 1.19·11-s + (−0.782 − 1.35i)12-s + (−0.182 + 0.316i)13-s + 0.253·14-s + (−0.410 − 0.710i)15-s + (−0.457 − 0.791i)16-s + (0.375 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.388 + 0.921i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.388 + 0.921i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.13644 - 1.71300i\)
\(L(\frac12)\) \(\approx\) \(1.13644 - 1.71300i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (5.94e3 + 5.83e3i)T \)
good2 \( 1 + (0.480 + 0.832i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-12.5 + 21.7i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-14.2 + 24.6i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (96.5 - 167. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 478.T + 1.61e5T^{2} \)
13 \( 1 + (111. - 192. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-447. - 775. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.00e3 + 1.74e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 2.89e3T + 6.43e6T^{2} \)
29 \( 1 - 2.82e3T + 2.05e7T^{2} \)
31 \( 1 - 4.93e3T + 2.86e7T^{2} \)
41 \( 1 + (1.59e3 - 2.76e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 1.90e4T + 1.47e8T^{2} \)
47 \( 1 + 2.58e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.39e4 - 2.41e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.28e4 - 2.21e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-5.72e3 + 9.90e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (4.82e3 - 8.36e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (1.75e4 - 3.03e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 - 3.99e3T + 2.07e9T^{2} \)
79 \( 1 + (3.25e4 - 5.63e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-5.65e3 - 9.80e3i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (5.31e4 + 9.20e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 7.97e4T + 8.58e9T^{2} \)
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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

−14.76744679680486132812429232093, −13.81091244628994248942131122878, −12.50617503548367272524371786398, −11.76665128259323113035320692616, −9.564255729630501492265250015535, −8.726330848024209195041945398672, −6.88805717449843450927621264887, −5.89649164197559826160819169684, −2.61891687421467904953661419798, −1.30493798374440984877377866653, 3.15327846627316421437592083297, 4.04818224724072915340974730885, 6.67122510265840655657839813641, 8.116585907714335978568819598240, 9.666569164430574042790953066495, 10.38651664095541621818631846999, 11.98685594939267282107093489323, 13.78857556484194861169109886154, 14.53901874462197577140507244667, 16.03397869097998251239167863499

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