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Properties

Label 2-37-37.11-c5-0-8
Degree $2$
Conductor $37$
Sign $-0.424 + 0.905i$
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  + (−1.48 + 0.856i)2-s + (−9.17 + 15.8i)3-s + (−14.5 + 25.1i)4-s + (−29.0 − 16.7i)5-s − 31.4i·6-s + (80.8 − 139. i)7-s − 104. i·8-s + (−46.8 − 81.2i)9-s + 57.4·10-s − 121.·11-s + (−266. − 461. i)12-s + (−328. − 189. i)13-s + 277. i·14-s + (532. − 307. i)15-s + (−375. − 650. i)16-s + (−886. + 511. i)17-s + ⋯
L(s)  = 1  + (−0.262 + 0.151i)2-s + (−0.588 + 1.01i)3-s + (−0.454 + 0.786i)4-s + (−0.519 − 0.299i)5-s − 0.356i·6-s + (0.623 − 1.07i)7-s − 0.578i·8-s + (−0.192 − 0.334i)9-s + 0.181·10-s − 0.302·11-s + (−0.534 − 0.925i)12-s + (−0.539 − 0.311i)13-s + 0.377i·14-s + (0.611 − 0.352i)15-s + (−0.366 − 0.634i)16-s + (−0.743 + 0.429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.424 + 0.905i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.424 + 0.905i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0128382 - 0.0202079i\)
\(L(\frac12)\) \(\approx\) \(0.0128382 - 0.0202079i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (8.14e3 + 1.73e3i)T \)
good2 \( 1 + (1.48 - 0.856i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (9.17 - 15.8i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (29.0 + 16.7i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-80.8 + 139. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 121.T + 1.61e5T^{2} \)
13 \( 1 + (328. + 189. i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (886. - 511. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (372. + 215. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 49.1iT - 6.43e6T^{2} \)
29 \( 1 - 2.13e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.02e3iT - 2.86e7T^{2} \)
41 \( 1 + (-2.73e3 + 4.73e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + 1.12e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.72e4T + 2.29e8T^{2} \)
53 \( 1 + (-5.16e3 - 8.95e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (4.08e4 - 2.35e4i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.24e4 - 1.29e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (5.42e3 - 9.38e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (2.25e4 - 3.89e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 - 8.88e4T + 2.07e9T^{2} \)
79 \( 1 + (4.92e4 + 2.84e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-5.36e4 - 9.29e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (4.33e4 - 2.50e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.54e5iT - 8.58e9T^{2} \)
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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

−15.36885296260225455153911747503, −13.78119947363716887891454024223, −12.46338791100113612285546628472, −11.08819932812789859848680641351, −10.08258630364354549846361673621, −8.517587662845858449708613575886, −7.33151298308419670537464917905, −4.85179922675633925614224228795, −3.96058941819702073016300535011, −0.01603132155548028371047715666, 1.91079272935681608878942180416, 5.04049382684501524182560144369, 6.41814189468327679727764388131, 7.989824082104543309534217997880, 9.430541808093653691756533654327, 11.15087477473960254368710704902, 11.90635301588277446692829141848, 13.22295212090431257552124597057, 14.62912678718899598230931434817, 15.51822900468315309687695219381

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