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Properties

Label 2-37-37.10-c7-0-1
Degree $2$
Conductor $37$
Sign $0.0856 - 0.996i$
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  + (−8.90 + 15.4i)2-s + (0.464 + 0.804i)3-s + (−94.5 − 163. i)4-s + (−251. − 434. i)5-s − 16.5·6-s + (772. + 1.33e3i)7-s + 1.08e3·8-s + (1.09e3 − 1.89e3i)9-s + 8.94e3·10-s − 1.20e3·11-s + (87.8 − 152. i)12-s + (4.18e3 + 7.24e3i)13-s − 2.75e4·14-s + (233. − 404. i)15-s + (2.40e3 − 4.17e3i)16-s + (−135. + 234. i)17-s + ⋯
L(s)  = 1  + (−0.787 + 1.36i)2-s + (0.00993 + 0.0172i)3-s + (−0.738 − 1.27i)4-s + (−0.898 − 1.55i)5-s − 0.0312·6-s + (0.851 + 1.47i)7-s + 0.751·8-s + (0.499 − 0.865i)9-s + 2.82·10-s − 0.273·11-s + (0.0146 − 0.0254i)12-s + (0.528 + 0.914i)13-s − 2.67·14-s + (0.0178 − 0.0309i)15-s + (0.147 − 0.254i)16-s + (−0.00667 + 0.0115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.0856 - 0.996i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.0856 - 0.996i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.753740 + 0.691746i\)
\(L(\frac12)\) \(\approx\) \(0.753740 + 0.691746i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.95e5 + 8.83e4i)T \)
good2 \( 1 + (8.90 - 15.4i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (-0.464 - 0.804i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (251. + 434. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-772. - 1.33e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + 1.20e3T + 1.94e7T^{2} \)
13 \( 1 + (-4.18e3 - 7.24e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (135. - 234. i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-6.91e3 - 1.19e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 4.34e4T + 3.40e9T^{2} \)
29 \( 1 - 1.74e5T + 1.72e10T^{2} \)
31 \( 1 + 7.74e4T + 2.75e10T^{2} \)
41 \( 1 + (-3.24e4 - 5.61e4i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 9.07e5T + 2.71e11T^{2} \)
47 \( 1 - 8.26e5T + 5.06e11T^{2} \)
53 \( 1 + (6.74e5 - 1.16e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.11e6 + 1.93e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-7.94e5 - 1.37e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (8.70e5 + 1.50e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (5.03e5 + 8.71e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 3.16e6T + 1.10e13T^{2} \)
79 \( 1 + (2.51e6 + 4.35e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (1.74e6 - 3.02e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (-3.00e6 + 5.20e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 1.39e7T + 8.07e13T^{2} \)
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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

−15.66305759927187558058878489405, −14.56889995899347960002549126645, −12.55724028011928897172650705393, −11.72495060664597989408823155733, −9.147158638705013694382468068809, −8.776714900856164976385756224786, −7.66219364354741827907186603292, −5.88882573066652586323099102748, −4.60938477696674636210083456543, −1.01431921672327781810348467811, 0.850773308572338622982627362294, 2.76428824164156679222458292143, 4.10914141923768322472940862660, 7.26655073875845531182677152887, 8.073368661133695970165157430209, 10.34803490807995941349930279149, 10.71210168829350814697720770843, 11.44912058973332460149649839623, 13.16692650206629667789741016545, 14.41125278238724434025090952648

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