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Properties

Label 2-37-37.26-c3-0-2
Degree $2$
Conductor $37$
Sign $-0.340 - 0.940i$
Analytic cond. $2.18307$
Root an. cond. $1.47752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 + 3.25i)2-s + (−1.14 + 1.97i)3-s + (−3.04 + 5.27i)4-s + (−6.80 + 11.7i)5-s − 8.57·6-s + (13.2 − 22.9i)7-s + 7.14·8-s + (10.8 + 18.8i)9-s − 51.0·10-s − 21.7·11-s + (−6.96 − 12.0i)12-s + (26.5 − 45.9i)13-s + 99.4·14-s + (−15.5 − 26.9i)15-s + (37.8 + 65.4i)16-s + (1.43 + 2.49i)17-s + ⋯
L(s)  = 1  + (0.663 + 1.14i)2-s + (−0.219 + 0.380i)3-s + (−0.381 + 0.659i)4-s + (−0.608 + 1.05i)5-s − 0.583·6-s + (0.714 − 1.23i)7-s + 0.315·8-s + (0.403 + 0.698i)9-s − 1.61·10-s − 0.596·11-s + (−0.167 − 0.290i)12-s + (0.565 − 0.979i)13-s + 1.89·14-s + (−0.267 − 0.463i)15-s + (0.590 + 1.02i)16-s + (0.0205 + 0.0355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ -0.340 - 0.940i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.937958 + 1.33740i\)
\(L(\frac12)\) \(\approx\) \(0.937958 + 1.33740i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (224. + 6.51i)T \)
good2 \( 1 + (-1.87 - 3.25i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (1.14 - 1.97i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (6.80 - 11.7i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-13.2 + 22.9i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + 21.7T + 1.33e3T^{2} \)
13 \( 1 + (-26.5 + 45.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-1.43 - 2.49i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-56.5 + 97.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 82.1T + 1.21e4T^{2} \)
29 \( 1 + 239.T + 2.43e4T^{2} \)
31 \( 1 + 34.4T + 2.97e4T^{2} \)
41 \( 1 + (-90.1 + 156. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 224.T + 7.95e4T^{2} \)
47 \( 1 - 420.T + 1.03e5T^{2} \)
53 \( 1 + (97.5 + 169. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (44.3 + 76.7i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (334. - 579. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (428. - 742. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-512. + 887. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 1.06e3T + 3.89e5T^{2} \)
79 \( 1 + (152. - 264. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (4.37 + 7.57i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (494. + 856. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.73e3T + 9.12e5T^{2} \)
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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

−15.85602353305866892057745155606, −15.27370534691943938438445772748, −14.08173448450459352869873093821, −13.27918078164331233135023334169, −10.96544498708468261885912443580, −10.56393602736857192083734076027, −7.73092822509402910644773943740, −7.25248446669986630191784934168, −5.37447212923060233356877676789, −3.97017610186711986004538071785, 1.67168871830167859027954056224, 4.01396704034899795264502376457, 5.49894031451699759050101162015, 7.900696024068617622826026801720, 9.322289488463679301845738634613, 11.22591622997205563160134781066, 12.19486194548420268049192363967, 12.50012027302179241609970445747, 13.97014071350870921819654002557, 15.48081228949424334415375018255

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