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Properties

Label 2-37-37.36-c5-0-1
Degree $2$
Conductor $37$
Sign $0.452 + 0.891i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.0i·2-s − 25.7·3-s − 69.2·4-s + 77.4i·5-s − 259. i·6-s + 168.·7-s − 374. i·8-s + 420.·9-s − 778.·10-s − 603.·11-s + 1.78e3·12-s − 268. i·13-s + 1.69e3i·14-s − 1.99e3i·15-s + 1.55e3·16-s + 873. i·17-s + ⋯
L(s)  = 1  + 1.77i·2-s − 1.65·3-s − 2.16·4-s + 1.38i·5-s − 2.93i·6-s + 1.29·7-s − 2.06i·8-s + 1.73·9-s − 2.46·10-s − 1.50·11-s + 3.57·12-s − 0.440i·13-s + 2.30i·14-s − 2.28i·15-s + 1.51·16-s + 0.733i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.452 + 0.891i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.452 + 0.891i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.290160 - 0.178125i\)
\(L(\frac12)\) \(\approx\) \(0.290160 - 0.178125i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (3.76e3 + 7.42e3i)T \)
good2 \( 1 - 10.0iT - 32T^{2} \)
3 \( 1 + 25.7T + 243T^{2} \)
5 \( 1 - 77.4iT - 3.12e3T^{2} \)
7 \( 1 - 168.T + 1.68e4T^{2} \)
11 \( 1 + 603.T + 1.61e5T^{2} \)
13 \( 1 + 268. iT - 3.71e5T^{2} \)
17 \( 1 - 873. iT - 1.41e6T^{2} \)
19 \( 1 + 706. iT - 2.47e6T^{2} \)
23 \( 1 + 141. iT - 6.43e6T^{2} \)
29 \( 1 - 6.35e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.75e3iT - 2.86e7T^{2} \)
41 \( 1 + 4.15e3T + 1.15e8T^{2} \)
43 \( 1 - 8.99e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.22e4T + 2.29e8T^{2} \)
53 \( 1 + 1.33e4T + 4.18e8T^{2} \)
59 \( 1 + 7.56e3iT - 7.14e8T^{2} \)
61 \( 1 - 5.37e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.07e3T + 1.35e9T^{2} \)
71 \( 1 - 4.15e4T + 1.80e9T^{2} \)
73 \( 1 + 5.06e4T + 2.07e9T^{2} \)
79 \( 1 - 9.48e3iT - 3.07e9T^{2} \)
83 \( 1 - 1.47e3T + 3.93e9T^{2} \)
89 \( 1 + 1.11e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.75e4iT - 8.58e9T^{2} \)
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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

−16.30140330016564574836594192863, −15.29195914540284742268716297374, −14.53067731032954608474459657178, −13.02573798640195492231397127413, −11.18887467844422174157706528876, −10.41944934767781002325440277665, −7.985172085012892115430471518638, −7.00663486758224354629878227326, −5.80112787279832768683331694921, −4.88715286362722921129348849333, 0.25403096813578289458820804637, 1.56922129396027014358999837069, 4.72453538671751119996328499557, 5.15159210128458747084348448948, 8.269776713086005090645590285895, 9.920290177818404624463960635081, 11.02419238840894107072389970016, 11.80934031036697625946545908427, 12.55722838940196574931467117395, 13.61701305598011640843519122780

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