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Properties

Label 2-37-37.26-c1-0-0
Degree $2$
Conductor $37$
Sign $0.729 + 0.683i$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1 + 1.73i)7-s − 3·8-s + (1.5 + 2.59i)9-s + 0.999·10-s − 2·11-s + (1 − 1.73i)13-s + 1.99·14-s + (0.500 + 0.866i)16-s + (−1.5 − 2.59i)17-s + (1.5 − 2.59i)18-s + (3 − 5.19i)19-s + (0.499 + 0.866i)20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.377 + 0.654i)7-s − 1.06·8-s + (0.5 + 0.866i)9-s + 0.316·10-s − 0.603·11-s + (0.277 − 0.480i)13-s + 0.534·14-s + (0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + (0.353 − 0.612i)18-s + (0.688 − 1.19i)19-s + (0.111 + 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.729 + 0.683i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602131 - 0.238056i\)
\(L(\frac12)\) \(\approx\) \(0.602131 - 0.238056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (5.5 - 2.59i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−15.89959470749858249320840417770, −15.49686103420560485601506520234, −13.88291242818527276459679165745, −12.50740797366944525930206561141, −11.18579217816029458093362187140, −10.32278704395388493881457457093, −8.997337969576225080178847698983, −7.18371500213717680724324414889, −5.42454872665064942861452531044, −2.67162594208091206392648495769, 3.81870136798848822668749225626, 6.27178619007762406330651043689, 7.51862835549577123370251104304, 8.778829562106292447356049750462, 10.23218905869563274534139853984, 11.98519994974156085879271117347, 12.86305075480989267822374846899, 14.45606531955039221100997664345, 15.92126559659218090507148212754, 16.30070774625329966321574399972

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