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Properties

Label 2-37-37.26-c5-0-8
Degree $2$
Conductor $37$
Sign $0.993 + 0.117i$
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  + (1.06 + 1.85i)2-s + (−4.46 + 7.73i)3-s + (13.7 − 23.7i)4-s + (22.0 − 38.1i)5-s − 19.1·6-s + (29.8 − 51.7i)7-s + 127.·8-s + (81.6 + 141. i)9-s + 94.1·10-s + 415.·11-s + (122. + 212. i)12-s + (102. − 177. i)13-s + 127.·14-s + (196. + 340. i)15-s + (−302. − 524. i)16-s + (−311. − 539. i)17-s + ⋯
L(s)  = 1  + (0.189 + 0.327i)2-s + (−0.286 + 0.495i)3-s + (0.428 − 0.742i)4-s + (0.393 − 0.681i)5-s − 0.216·6-s + (0.230 − 0.399i)7-s + 0.702·8-s + (0.335 + 0.581i)9-s + 0.297·10-s + 1.03·11-s + (0.245 + 0.425i)12-s + (0.167 − 0.290i)13-s + 0.174·14-s + (0.225 + 0.390i)15-s + (−0.295 − 0.512i)16-s + (−0.261 − 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.993 + 0.117i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.993 + 0.117i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.98180 - 0.116914i\)
\(L(\frac12)\) \(\approx\) \(1.98180 - 0.116914i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (3.95e3 - 7.33e3i)T \)
good2 \( 1 + (-1.06 - 1.85i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (4.46 - 7.73i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-22.0 + 38.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-29.8 + 51.7i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 415.T + 1.61e5T^{2} \)
13 \( 1 + (-102. + 177. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (311. + 539. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-26.8 + 46.5i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 - 62.5T + 6.43e6T^{2} \)
29 \( 1 + 2.78e3T + 2.05e7T^{2} \)
31 \( 1 + 5.10e3T + 2.86e7T^{2} \)
41 \( 1 + (3.31e3 - 5.74e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 7.37e3T + 1.47e8T^{2} \)
47 \( 1 - 1.05e4T + 2.29e8T^{2} \)
53 \( 1 + (14.3 + 24.7i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (9.23e3 + 1.60e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.66e4 - 4.60e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.06e3 - 3.56e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (3.48e4 - 6.03e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 - 1.74e4T + 2.07e9T^{2} \)
79 \( 1 + (6.39e3 - 1.10e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (5.43e4 + 9.40e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (1.19e4 + 2.06e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.34e5T + 8.58e9T^{2} \)
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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

−15.43930505016331744217645092483, −14.25259651570097179583146802043, −13.19145377022440070965297247227, −11.45219121638368322515465065031, −10.40095668263235860073971226073, −9.203164366298137512296002391297, −7.25830890009939835173216317502, −5.68274136402931426279324295120, −4.51408005955994054102383905700, −1.42384242410431886222579486144, 1.90606866313396266996538957042, 3.77533276570437928384581367521, 6.25885715557440840408309080760, 7.26780524787217028999302132089, 9.039012401229317734083719333779, 10.79703098925590565531794030233, 11.87120303567605359711813376335, 12.70933815563226044337207345190, 14.07562141715824045750565276966, 15.29714711887609162476476439602

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