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Properties

Label 2-37-37.6-c8-0-18
Degree $2$
Conductor $37$
Sign $-0.608 + 0.793i$
Analytic cond. $15.0730$
Root an. cond. $3.88239$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.8 + 13.8i)2-s − 94.2i·3-s − 129. i·4-s + (−267. − 267. i)5-s + (1.30e3 + 1.30e3i)6-s + 3.63e3·7-s + (−1.75e3 − 1.75e3i)8-s − 2.32e3·9-s + 7.42e3·10-s + 1.85e4i·11-s − 1.21e4·12-s + (−3.18e4 − 3.18e4i)13-s + (−5.05e4 + 5.05e4i)14-s + (−2.51e4 + 2.51e4i)15-s + 8.19e4·16-s + (3.92e4 + 3.92e4i)17-s + ⋯
L(s)  = 1  + (−0.867 + 0.867i)2-s − 1.16i·3-s − 0.505i·4-s + (−0.427 − 0.427i)5-s + (1.00 + 1.00i)6-s + 1.51·7-s + (−0.429 − 0.429i)8-s − 0.354·9-s + 0.742·10-s + 1.26i·11-s − 0.587·12-s + (−1.11 − 1.11i)13-s + (−1.31 + 1.31i)14-s + (−0.497 + 0.497i)15-s + 1.24·16-s + (0.470 + 0.470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ -0.608 + 0.793i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.235513 - 0.477231i\)
\(L(\frac12)\) \(\approx\) \(0.235513 - 0.477231i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.83e6 - 3.98e5i)T \)
good2 \( 1 + (13.8 - 13.8i)T - 256iT^{2} \)
3 \( 1 + 94.2iT - 6.56e3T^{2} \)
5 \( 1 + (267. + 267. i)T + 3.90e5iT^{2} \)
7 \( 1 - 3.63e3T + 5.76e6T^{2} \)
11 \( 1 - 1.85e4iT - 2.14e8T^{2} \)
13 \( 1 + (3.18e4 + 3.18e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-3.92e4 - 3.92e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (9.71e4 + 9.71e4i)T + 1.69e10iT^{2} \)
23 \( 1 + (2.64e5 + 2.64e5i)T + 7.83e10iT^{2} \)
29 \( 1 + (5.71e4 - 5.71e4i)T - 5.00e11iT^{2} \)
31 \( 1 + (7.19e5 - 7.19e5i)T - 8.52e11iT^{2} \)
41 \( 1 - 3.88e6iT - 7.98e12T^{2} \)
43 \( 1 + (4.23e6 + 4.23e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 3.43e6T + 2.38e13T^{2} \)
53 \( 1 + 7.15e6T + 6.22e13T^{2} \)
59 \( 1 + (-3.31e6 - 3.31e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (-1.15e7 + 1.15e7i)T - 1.91e14iT^{2} \)
67 \( 1 + 1.07e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.06e7T + 6.45e14T^{2} \)
73 \( 1 + 3.14e7iT - 8.06e14T^{2} \)
79 \( 1 + (-4.90e7 - 4.90e7i)T + 1.51e15iT^{2} \)
83 \( 1 - 2.50e7T + 2.25e15T^{2} \)
89 \( 1 + (-1.38e7 + 1.38e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-3.11e6 - 3.11e6i)T + 7.83e15iT^{2} \)
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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

−14.56779042871542113717267695366, −12.65507036428911689312365704484, −12.14986396096443152197813902202, −10.18323611594701190622418982604, −8.328678786482404201154256615545, −7.83225224313182538726167248545, −6.81645368770605646476270315822, −4.85087842146756612985797075924, −1.84341062266485524440247994914, −0.28147833704327056696933057264, 1.80099631524124496525971192193, 3.70263620754267486401883939771, 5.26504611236297689742246921092, 7.82999765048368189120392608274, 9.074744100027800597495437788145, 10.18043233551470341641840898626, 11.22305378433489371814514232256, 11.72659570502051393409860681373, 14.29654989076859293857378128548, 14.89187170025532944931360407820

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