Properties

Label 2-37-37.29-c4-0-7
Degree $2$
Conductor $37$
Sign $0.175 + 0.984i$
Analytic cond. $3.82468$
Root an. cond. $1.95568$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.76 + 6.59i)2-s + (−1.46 − 0.844i)3-s + (−26.5 − 15.3i)4-s + (−9.89 − 36.9i)5-s + (8.16 − 8.16i)6-s + (−22.6 + 39.1i)7-s + (70.9 − 70.9i)8-s + (−39.0 − 67.6i)9-s + 261.·10-s + 169. i·11-s + (25.9 + 44.8i)12-s + (−58.7 − 219. i)13-s + (−218. − 218. i)14-s + (−16.7 + 62.3i)15-s + (97.2 + 168. i)16-s + (−335. − 89.9i)17-s + ⋯
L(s)  = 1  + (−0.442 + 1.64i)2-s + (−0.162 − 0.0938i)3-s + (−1.66 − 0.958i)4-s + (−0.395 − 1.47i)5-s + (0.226 − 0.226i)6-s + (−0.461 + 0.799i)7-s + (1.10 − 1.10i)8-s + (−0.482 − 0.835i)9-s + 2.61·10-s + 1.40i·11-s + (0.179 + 0.311i)12-s + (−0.347 − 1.29i)13-s + (−1.11 − 1.11i)14-s + (−0.0742 + 0.277i)15-s + (0.380 + 0.658i)16-s + (−1.16 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(3.82468\)
Root analytic conductor: \(1.95568\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :2),\ 0.175 + 0.984i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.129420 - 0.108360i\)
\(L(\frac12)\) \(\approx\) \(0.129420 - 0.108360i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (848. + 1.07e3i)T \)
good2 \( 1 + (1.76 - 6.59i)T + (-13.8 - 8i)T^{2} \)
3 \( 1 + (1.46 + 0.844i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (9.89 + 36.9i)T + (-541. + 312.5i)T^{2} \)
7 \( 1 + (22.6 - 39.1i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 - 169. iT - 1.46e4T^{2} \)
13 \( 1 + (58.7 + 219. i)T + (-2.47e4 + 1.42e4i)T^{2} \)
17 \( 1 + (335. + 89.9i)T + (7.23e4 + 4.17e4i)T^{2} \)
19 \( 1 + (-41.1 - 153. i)T + (-1.12e5 + 6.51e4i)T^{2} \)
23 \( 1 + (150. - 150. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-486. - 486. i)T + 7.07e5iT^{2} \)
31 \( 1 + (784. + 784. i)T + 9.23e5iT^{2} \)
41 \( 1 + (-1.92e3 - 1.11e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (815. - 815. i)T - 3.41e6iT^{2} \)
47 \( 1 + 1.04e3T + 4.87e6T^{2} \)
53 \( 1 + (-576. - 998. i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (843. + 226. i)T + (1.04e7 + 6.05e6i)T^{2} \)
61 \( 1 + (-4.59e3 + 1.23e3i)T + (1.19e7 - 6.92e6i)T^{2} \)
67 \( 1 + (6.70e3 + 3.87e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-57.8 + 100. i)T + (-1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + 9.67e3iT - 2.83e7T^{2} \)
79 \( 1 + (2.65e3 + 9.91e3i)T + (-3.37e7 + 1.94e7i)T^{2} \)
83 \( 1 + (173. + 300. i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (927. - 3.45e3i)T + (-5.43e7 - 3.13e7i)T^{2} \)
97 \( 1 + (-7.31e3 + 7.31e3i)T - 8.85e7iT^{2} \)
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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

−15.53218261725732905842942695534, −14.84506058558923939399404864379, −12.95277166370953272469756475513, −12.16896735940207937992917427459, −9.529813456752715136898275153100, −8.799449534954338331416002883374, −7.55917483127144589302499335416, −5.97649120129267546696695632241, −4.82812975809646631530272946443, −0.13158494003155234244077702891, 2.59105428350386759062861932674, 3.92851349910921797507478472935, 6.80249431149971001425339874059, 8.634239902924681020502892773336, 10.25266921378367597923043528470, 11.00975533873772742720659823691, 11.58730052883332751283139326659, 13.44526357283116335195755961665, 14.17217459665424414780402311277, 16.12533723175881912582866755646

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