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Properties

Label 2-37-37.36-c7-0-8
Degree $2$
Conductor $37$
Sign $-0.402 - 0.915i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.3i·2-s + 76.2·3-s − 25.5·4-s + 93.6i·5-s + 944. i·6-s − 1.23e3·7-s + 1.26e3i·8-s + 3.62e3·9-s − 1.15e3·10-s + 6.53e3·11-s − 1.94e3·12-s + 1.41e4i·13-s − 1.53e4i·14-s + 7.13e3i·15-s − 1.89e4·16-s − 1.41e4i·17-s + ⋯
L(s)  = 1  + 1.09i·2-s + 1.63·3-s − 0.199·4-s + 0.334i·5-s + 1.78i·6-s − 1.36·7-s + 0.876i·8-s + 1.65·9-s − 0.366·10-s + 1.47·11-s − 0.324·12-s + 1.78i·13-s − 1.49i·14-s + 0.545i·15-s − 1.15·16-s − 0.696i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.402 - 0.915i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.402 - 0.915i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.62801 + 2.49525i\)
\(L(\frac12)\) \(\approx\) \(1.62801 + 2.49525i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.24e5 + 2.82e5i)T \)
good2 \( 1 - 12.3iT - 128T^{2} \)
3 \( 1 - 76.2T + 2.18e3T^{2} \)
5 \( 1 - 93.6iT - 7.81e4T^{2} \)
7 \( 1 + 1.23e3T + 8.23e5T^{2} \)
11 \( 1 - 6.53e3T + 1.94e7T^{2} \)
13 \( 1 - 1.41e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.41e4iT - 4.10e8T^{2} \)
19 \( 1 + 2.73e4iT - 8.93e8T^{2} \)
23 \( 1 - 4.21e4iT - 3.40e9T^{2} \)
29 \( 1 + 2.45e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.24e4iT - 2.75e10T^{2} \)
41 \( 1 + 1.63e5T + 1.94e11T^{2} \)
43 \( 1 + 6.18e5iT - 2.71e11T^{2} \)
47 \( 1 - 4.84e5T + 5.06e11T^{2} \)
53 \( 1 - 9.17e4T + 1.17e12T^{2} \)
59 \( 1 - 1.84e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.40e6iT - 3.14e12T^{2} \)
67 \( 1 - 4.56e6T + 6.06e12T^{2} \)
71 \( 1 + 1.11e6T + 9.09e12T^{2} \)
73 \( 1 - 1.33e6T + 1.10e13T^{2} \)
79 \( 1 + 4.36e6iT - 1.92e13T^{2} \)
83 \( 1 + 4.11e5T + 2.71e13T^{2} \)
89 \( 1 - 1.40e6iT - 4.42e13T^{2} \)
97 \( 1 - 9.97e6iT - 8.07e13T^{2} \)
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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

−15.18742127437767067286633588904, −14.16024154573023432738206963937, −13.60560514640038351028418026922, −11.71273439046686899981378220802, −9.434658084925467439928641803074, −8.943793063598157185322547879241, −7.18656109270948807428579854488, −6.57177283766865381099643534121, −3.91357359062663640003241039725, −2.36864845340846375173578919762, 1.25812920343296755785578791512, 2.99906624304700731977278174477, 3.66390108521890125117000157857, 6.69474538377018290753002855916, 8.467189074961036474827888385084, 9.542918501100957561555076316604, 10.44430447302425940253363613506, 12.56925914500248453964650946983, 12.83779312500540622482464491255, 14.33242869395241130739946986548

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