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Properties

Label 2-37-37.26-c5-0-10
Degree $2$
Conductor $37$
Sign $0.980 - 0.196i$
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  + (3.71 + 6.43i)2-s + (6.84 − 11.8i)3-s + (−11.5 + 20.0i)4-s + (46.2 − 80.1i)5-s + 101.·6-s + (3.53 − 6.13i)7-s + 65.8·8-s + (27.7 + 48.0i)9-s + 687.·10-s − 602.·11-s + (158. + 274. i)12-s + (−118. + 205. i)13-s + 52.5·14-s + (−633. − 1.09e3i)15-s + (614. + 1.06e3i)16-s + (917. + 1.58e3i)17-s + ⋯
L(s)  = 1  + (0.656 + 1.13i)2-s + (0.439 − 0.760i)3-s + (−0.361 + 0.625i)4-s + (0.827 − 1.43i)5-s + 1.15·6-s + (0.0273 − 0.0472i)7-s + 0.363·8-s + (0.114 + 0.197i)9-s + 2.17·10-s − 1.50·11-s + (0.317 + 0.549i)12-s + (−0.194 + 0.336i)13-s + 0.0716·14-s + (−0.727 − 1.25i)15-s + (0.600 + 1.03i)16-s + (0.770 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.980 - 0.196i$
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: \((2,\ 37,\ (\ :5/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.74435 + 0.272200i\)
\(L(\frac12)\) \(\approx\) \(2.74435 + 0.272200i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.48e3 - 8.19e3i)T \)
good2 \( 1 + (-3.71 - 6.43i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-6.84 + 11.8i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-46.2 + 80.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-3.53 + 6.13i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 602.T + 1.61e5T^{2} \)
13 \( 1 + (118. - 205. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-917. - 1.58e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-488. + 846. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 1.08e3T + 6.43e6T^{2} \)
29 \( 1 + 6.09e3T + 2.05e7T^{2} \)
31 \( 1 - 2.22e3T + 2.86e7T^{2} \)
41 \( 1 + (-5.45e3 + 9.45e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + 1.72e4T + 1.47e8T^{2} \)
47 \( 1 + 4.13e3T + 2.29e8T^{2} \)
53 \( 1 + (-7.04e3 - 1.21e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (981. + 1.69e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-3.54e3 + 6.13e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-8.45e3 + 1.46e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-3.02e4 + 5.24e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + 7.25e4T + 2.07e9T^{2} \)
79 \( 1 + (2.68e4 - 4.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.54e4 + 2.67e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (4.65e4 + 8.06e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 6.48e4T + 8.58e9T^{2} \)
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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

−15.41345468441317753126447458836, −13.97772871427308864156929499268, −13.21796796396184796470763195392, −12.65452167806116356628686295479, −10.21585129438658237484923517096, −8.452794798839987827924362511686, −7.54581289997482941886797007424, −5.84019727077535427371193242651, −4.83029291995343901633434364549, −1.70578709741220113925246129544, 2.49403869729738931085602181304, 3.40850194720382299147550554515, 5.34504069933453347751453512318, 7.44569124535407316941985743930, 9.894023010511621690894461081555, 10.25145914272569615359437516402, 11.49230981994913431155711240615, 13.02061020349127527013169242750, 14.06648155729238308175305472057, 14.90100060599337824946383635295

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