Properties

Label 2-37-37.11-c7-0-15
Degree $2$
Conductor $37$
Sign $-0.873 + 0.486i$
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  + (−2.67 + 1.54i)2-s + (32.7 − 56.7i)3-s + (−59.2 + 102. i)4-s + (−63.8 − 36.8i)5-s + 202. i·6-s + (457. − 791. i)7-s − 761. i·8-s + (−1.05e3 − 1.81e3i)9-s + 227.·10-s − 5.54e3·11-s + (3.87e3 + 6.71e3i)12-s + (−810. − 468. i)13-s + 2.82e3i·14-s + (−4.17e3 + 2.41e3i)15-s + (−6.40e3 − 1.10e4i)16-s + (−1.32e4 + 7.66e3i)17-s + ⋯
L(s)  = 1  + (−0.236 + 0.136i)2-s + (0.700 − 1.21i)3-s + (−0.462 + 0.801i)4-s + (−0.228 − 0.131i)5-s + 0.382i·6-s + (0.503 − 0.872i)7-s − 0.526i·8-s + (−0.480 − 0.831i)9-s + 0.0720·10-s − 1.25·11-s + (0.647 + 1.12i)12-s + (−0.102 − 0.0590i)13-s + 0.275i·14-s + (−0.319 + 0.184i)15-s + (−0.390 − 0.676i)16-s + (−0.655 + 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.873 + 0.486i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.873 + 0.486i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.230491 - 0.888606i\)
\(L(\frac12)\) \(\approx\) \(0.230491 - 0.888606i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.07e5 - 2.28e5i)T \)
good2 \( 1 + (2.67 - 1.54i)T + (64 - 110. i)T^{2} \)
3 \( 1 + (-32.7 + 56.7i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (63.8 + 36.8i)T + (3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-457. + 791. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 5.54e3T + 1.94e7T^{2} \)
13 \( 1 + (810. + 468. i)T + (3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (1.32e4 - 7.66e3i)T + (2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (3.93e4 + 2.27e4i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + 1.53e4iT - 3.40e9T^{2} \)
29 \( 1 + 3.34e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.02e5iT - 2.75e10T^{2} \)
41 \( 1 + (-1.35e5 + 2.34e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + 8.11e5iT - 2.71e11T^{2} \)
47 \( 1 - 4.65e5T + 5.06e11T^{2} \)
53 \( 1 + (-6.77e5 - 1.17e6i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-1.72e6 + 9.94e5i)T + (1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.28e6 + 7.39e5i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.09e6 + 1.89e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-1.50e5 + 2.60e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + 3.94e6T + 1.10e13T^{2} \)
79 \( 1 + (-2.81e6 - 1.62e6i)T + (9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-3.52e5 - 6.10e5i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (8.88e6 - 5.12e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.06e7iT - 8.07e13T^{2} \)
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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

−13.91984676383386730394935088388, −13.21699876939810386432710822403, −12.37829227699574929705216709429, −10.62315049290465873880629481438, −8.625373466752185597130069080385, −7.919490942628339129320090997133, −6.95778562468016785773207904068, −4.35481972983399353692589634459, −2.44012728920569295500111825849, −0.39046543192211702788823494161, 2.37404437253418149542225557938, 4.35012598507676964112724813759, 5.56786761989605095146971036389, 8.194586563671214810112866261024, 9.177782388980744682670497968346, 10.22416638204318760453171959156, 11.23418557247739496348551492826, 13.18566732805317851014382751074, 14.70057820571631074012079201087, 15.07988026290695332618094491022

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