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Properties

Label 2-37-37.10-c7-0-15
Degree $2$
Conductor $37$
Sign $-0.567 + 0.823i$
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  + (4.93 − 8.54i)2-s + (−6.61 − 11.4i)3-s + (15.2 + 26.4i)4-s + (56.3 + 97.5i)5-s − 130.·6-s + (−783. − 1.35e3i)7-s + 1.56e3·8-s + (1.00e3 − 1.74e3i)9-s + 1.11e3·10-s − 163.·11-s + (202. − 349. i)12-s + (−5.39e3 − 9.34e3i)13-s − 1.54e4·14-s + (745. − 1.29e3i)15-s + (5.77e3 − 9.99e3i)16-s + (−3.75e3 + 6.50e3i)17-s + ⋯
L(s)  = 1  + (0.436 − 0.755i)2-s + (−0.141 − 0.244i)3-s + (0.119 + 0.206i)4-s + (0.201 + 0.349i)5-s − 0.246·6-s + (−0.863 − 1.49i)7-s + 1.08·8-s + (0.460 − 0.796i)9-s + 0.351·10-s − 0.0370·11-s + (0.0337 − 0.0584i)12-s + (−0.681 − 1.18i)13-s − 1.50·14-s + (0.0569 − 0.0987i)15-s + (0.352 − 0.609i)16-s + (−0.185 + 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.940528 - 1.79122i\)
\(L(\frac12)\) \(\approx\) \(0.940528 - 1.79122i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (3.00e5 + 6.94e4i)T \)
good2 \( 1 + (-4.93 + 8.54i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (6.61 + 11.4i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-56.3 - 97.5i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (783. + 1.35e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + 163.T + 1.94e7T^{2} \)
13 \( 1 + (5.39e3 + 9.34e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (3.75e3 - 6.50e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (1.00e3 + 1.74e3i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + 1.16e4T + 3.40e9T^{2} \)
29 \( 1 - 3.05e3T + 1.72e10T^{2} \)
31 \( 1 - 2.30e5T + 2.75e10T^{2} \)
41 \( 1 + (-2.86e5 - 4.95e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 1.52e5T + 2.71e11T^{2} \)
47 \( 1 - 1.07e6T + 5.06e11T^{2} \)
53 \( 1 + (6.25e5 - 1.08e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-7.48e5 + 1.29e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-5.92e5 - 1.02e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.03e6 + 1.80e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (9.16e5 + 1.58e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 2.62e6T + 1.10e13T^{2} \)
79 \( 1 + (-3.05e6 - 5.29e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (1.35e6 - 2.35e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (5.68e6 - 9.84e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 4.11e6T + 8.07e13T^{2} \)
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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

−14.01632254356148228760012041677, −13.00421457362087342912344813563, −12.26908886400038645677278581148, −10.66689975605914621433457593661, −9.989128272065245120866436005740, −7.62274120622739594189576714162, −6.57274680534089424045030465589, −4.19099011688701771079302592933, −2.94921390973608884215781264183, −0.807887708976515933771135812877, 2.14359977068360604150430051035, 4.73353755324295332606792169441, 5.80890756489710638197889072548, 7.10345581278362232125615824125, 8.969039487498399047947510923167, 10.14845913685595935225722945410, 11.79590151622565812606394910731, 13.06311340387092882313382946222, 14.23536527745545041620790131398, 15.54415861022532393286397879364

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