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Properties

Label 2-37-37.36-c5-0-7
Degree $2$
Conductor $37$
Sign $-0.458 - 0.888i$
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  + 8.69i·2-s + 28.4·3-s − 43.5·4-s + 14.1i·5-s + 247. i·6-s + 47.5·7-s − 100. i·8-s + 567.·9-s − 123.·10-s − 461.·11-s − 1.24e3·12-s − 290. i·13-s + 413. i·14-s + 403. i·15-s − 518.·16-s − 1.76e3i·17-s + ⋯
L(s)  = 1  + 1.53i·2-s + 1.82·3-s − 1.36·4-s + 0.253i·5-s + 2.80i·6-s + 0.366·7-s − 0.557i·8-s + 2.33·9-s − 0.389·10-s − 1.15·11-s − 2.48·12-s − 0.476i·13-s + 0.563i·14-s + 0.463i·15-s − 0.506·16-s − 1.48i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.458 - 0.888i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.458 - 0.888i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.34756 + 2.21022i\)
\(L(\frac12)\) \(\approx\) \(1.34756 + 2.21022i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-3.81e3 - 7.40e3i)T \)
good2 \( 1 - 8.69iT - 32T^{2} \)
3 \( 1 - 28.4T + 243T^{2} \)
5 \( 1 - 14.1iT - 3.12e3T^{2} \)
7 \( 1 - 47.5T + 1.68e4T^{2} \)
11 \( 1 + 461.T + 1.61e5T^{2} \)
13 \( 1 + 290. iT - 3.71e5T^{2} \)
17 \( 1 + 1.76e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.71e3iT - 2.47e6T^{2} \)
23 \( 1 + 4.41e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.38e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.72e3iT - 2.86e7T^{2} \)
41 \( 1 - 2.51e3T + 1.15e8T^{2} \)
43 \( 1 + 1.75e4iT - 1.47e8T^{2} \)
47 \( 1 + 3.09e3T + 2.29e8T^{2} \)
53 \( 1 + 6.60e3T + 4.18e8T^{2} \)
59 \( 1 + 3.58e3iT - 7.14e8T^{2} \)
61 \( 1 + 2.46e3iT - 8.44e8T^{2} \)
67 \( 1 - 3.63e4T + 1.35e9T^{2} \)
71 \( 1 + 7.02e4T + 1.80e9T^{2} \)
73 \( 1 + 4.96e4T + 2.07e9T^{2} \)
79 \( 1 - 6.25e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.90e4T + 3.93e9T^{2} \)
89 \( 1 - 5.13e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.00e5iT - 8.58e9T^{2} \)
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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

−15.55855998663849661870705168170, −14.54788121029075336849288548140, −14.07493246035237273061088120542, −12.86718662333581468168127056021, −10.25045536129375369091262321356, −8.741539371025930383979416006424, −7.993383158734094196393565838267, −7.00395131900832388310570796424, −4.88052562542784418915245091763, −2.77622912387044383316522664036, 1.70805261014734585765396119325, 2.93338929676237187060563171373, 4.31467110855455199410699000238, 7.72635891793463787841269114950, 8.919044416198778620059775198200, 9.885627926663179000684030209065, 11.17619962077609283459786724491, 12.95208533617779985988393910799, 13.29789845700862997049961255937, 14.67168555702168895345932043696

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