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Properties

Label 2-37-37.28-c1-0-0
Degree $2$
Conductor $37$
Sign $0.287 - 0.957i$
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  + (−1.11 + 0.197i)2-s + (−0.522 + 2.96i)3-s + (−0.669 + 0.243i)4-s + (1.89 − 2.26i)5-s − 3.41i·6-s + (1.25 + 1.05i)7-s + (2.66 − 1.53i)8-s + (−5.67 − 2.06i)9-s + (−1.67 + 2.90i)10-s + (−0.588 − 1.01i)11-s + (−0.371 − 2.10i)12-s + (0.543 + 1.49i)13-s + (−1.61 − 0.929i)14-s + (5.71 + 6.80i)15-s + (−1.58 + 1.32i)16-s + (1.29 − 3.56i)17-s + ⋯
L(s)  = 1  + (−0.790 + 0.139i)2-s + (−0.301 + 1.70i)3-s + (−0.334 + 0.121i)4-s + (0.849 − 1.01i)5-s − 1.39i·6-s + (0.474 + 0.398i)7-s + (0.942 − 0.544i)8-s + (−1.89 − 0.688i)9-s + (−0.530 + 0.918i)10-s + (−0.177 − 0.307i)11-s + (−0.107 − 0.608i)12-s + (0.150 + 0.413i)13-s + (−0.430 − 0.248i)14-s + (1.47 + 1.75i)15-s + (−0.396 + 0.332i)16-s + (0.314 − 0.863i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.287 - 0.957i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408464 + 0.303968i\)
\(L(\frac12)\) \(\approx\) \(0.408464 + 0.303968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-3.80 - 4.74i)T \)
good2 \( 1 + (1.11 - 0.197i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (0.522 - 2.96i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-1.89 + 2.26i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.25 - 1.05i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.588 + 1.01i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.543 - 1.49i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.29 + 3.56i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (1.80 + 0.318i)T + (17.8 + 6.49i)T^{2} \)
23 \( 1 + (2.76 + 1.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.99 - 2.30i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.07iT - 31T^{2} \)
41 \( 1 + (0.914 - 0.332i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 0.234iT - 43T^{2} \)
47 \( 1 + (4.78 - 8.28i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.70 + 3.94i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-0.638 - 0.760i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.95 - 5.37i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-12.0 - 10.1i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.34 + 7.61i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 + (7.34 - 8.75i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (6.50 + 2.36i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (4.87 + 5.80i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-11.3 - 6.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−16.65974375587018847449908668076, −16.06475059784105628842081018800, −14.57898097052653729211769784207, −13.23241947174205715569662751501, −11.43969227866847611559732169355, −10.01930011219452885474430880982, −9.312637861832937073774607899142, −8.435018560849455313499805098590, −5.50021735919517053881495824705, −4.40878350334317995904115642845, 1.84340815313137188820669159024, 5.83479221692238965917945940821, 7.21872384596564351995844569676, 8.280796186301163152982255713002, 10.13929881773372788217912945518, 11.12532849394766990948439867255, 12.77531747610980739228690670112, 13.77446687854350858469045762112, 14.54954721604249445605286621200, 17.06691569206785068662751078410

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