Properties

Label 2-37-37.31-c6-0-10
Degree $2$
Conductor $37$
Sign $-0.421 + 0.906i$
Analytic cond. $8.51200$
Root an. cond. $2.91753$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.2 − 10.2i)2-s − 12.1i·3-s + 147. i·4-s + (92.7 − 92.7i)5-s + (−125. + 125. i)6-s + 599.·7-s + (862. − 862. i)8-s + 581.·9-s − 1.90e3·10-s + 1.23e3i·11-s + 1.79e3·12-s + (86.3 − 86.3i)13-s + (−6.16e3 − 6.16e3i)14-s + (−1.12e3 − 1.12e3i)15-s − 8.28e3·16-s + (3.85e3 − 3.85e3i)17-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)2-s − 0.450i·3-s + 2.30i·4-s + (0.741 − 0.741i)5-s + (−0.578 + 0.578i)6-s + 1.74·7-s + (1.68 − 1.68i)8-s + 0.797·9-s − 1.90·10-s + 0.931i·11-s + 1.03·12-s + (0.0392 − 0.0392i)13-s + (−2.24 − 2.24i)14-s + (−0.333 − 0.333i)15-s − 2.02·16-s + (0.784 − 0.784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.421 + 0.906i$
Analytic conductor: \(8.51200\)
Root analytic conductor: \(2.91753\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3),\ -0.421 + 0.906i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.666511 - 1.04459i\)
\(L(\frac12)\) \(\approx\) \(0.666511 - 1.04459i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.34e4 + 4.88e4i)T \)
good2 \( 1 + (10.2 + 10.2i)T + 64iT^{2} \)
3 \( 1 + 12.1iT - 729T^{2} \)
5 \( 1 + (-92.7 + 92.7i)T - 1.56e4iT^{2} \)
7 \( 1 - 599.T + 1.17e5T^{2} \)
11 \( 1 - 1.23e3iT - 1.77e6T^{2} \)
13 \( 1 + (-86.3 + 86.3i)T - 4.82e6iT^{2} \)
17 \( 1 + (-3.85e3 + 3.85e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (8.02e3 - 8.02e3i)T - 4.70e7iT^{2} \)
23 \( 1 + (-1.13e4 + 1.13e4i)T - 1.48e8iT^{2} \)
29 \( 1 + (-1.50e4 - 1.50e4i)T + 5.94e8iT^{2} \)
31 \( 1 + (1.19e4 + 1.19e4i)T + 8.87e8iT^{2} \)
41 \( 1 - 1.77e4iT - 4.75e9T^{2} \)
43 \( 1 + (4.05e4 - 4.05e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.86e4T + 1.07e10T^{2} \)
53 \( 1 + 2.39e5T + 2.21e10T^{2} \)
59 \( 1 + (-2.59e4 + 2.59e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (1.78e5 + 1.78e5i)T + 5.15e10iT^{2} \)
67 \( 1 - 5.04e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.10e5T + 1.28e11T^{2} \)
73 \( 1 - 3.05e4iT - 1.51e11T^{2} \)
79 \( 1 + (-4.12e5 + 4.12e5i)T - 2.43e11iT^{2} \)
83 \( 1 + 7.05e4T + 3.26e11T^{2} \)
89 \( 1 + (-3.99e4 - 3.99e4i)T + 4.96e11iT^{2} \)
97 \( 1 + (1.05e3 - 1.05e3i)T - 8.32e11iT^{2} \)
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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

−14.53383428235254079533539923477, −12.86485110018011384323831127894, −12.19500211689385529767889942569, −10.84774653578056045297279429938, −9.778117448632905791602083560936, −8.539890310878992948951465970898, −7.50909170475131489443184917936, −4.66588633622475672511168872126, −1.94691621431865171189956841074, −1.24468758788003789793965120050, 1.45847964998040280581123312217, 5.06316374622799605281278652495, 6.48874019833475745679072774795, 7.83981914651069153069851127109, 8.934223874311308706863137555961, 10.33223710850250783522948842576, 11.01351320083611314347787626195, 13.79899435108429812755371122451, 14.81351645615398696465952772936, 15.43827232471393460804072849382

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