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Properties

Label 2-37-37.27-c7-0-6
Degree $2$
Conductor $37$
Sign $-0.810 - 0.585i$
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  + (12.1 + 6.99i)2-s + (43.2 + 74.9i)3-s + (33.9 + 58.8i)4-s + (−221. + 127. i)5-s + 1.21e3i·6-s + (−32.7 − 56.6i)7-s − 840. i·8-s + (−2.65e3 + 4.59e3i)9-s − 3.57e3·10-s + 4.26e3·11-s + (−2.94e3 + 5.09e3i)12-s + (3.54e3 − 2.04e3i)13-s − 916. i·14-s + (−1.91e4 − 1.10e4i)15-s + (1.02e4 − 1.77e4i)16-s + (8.92e3 + 5.15e3i)17-s + ⋯
L(s)  = 1  + (1.07 + 0.618i)2-s + (0.925 + 1.60i)3-s + (0.265 + 0.459i)4-s + (−0.792 + 0.457i)5-s + 2.29i·6-s + (−0.0360 − 0.0624i)7-s − 0.580i·8-s + (−1.21 + 2.10i)9-s − 1.13·10-s + 0.966·11-s + (−0.491 + 0.850i)12-s + (0.447 − 0.258i)13-s − 0.0892i·14-s + (−1.46 − 0.846i)15-s + (0.624 − 1.08i)16-s + (0.440 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.810 - 0.585i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.810 - 0.585i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.07774 + 3.33011i\)
\(L(\frac12)\) \(\approx\) \(1.07774 + 3.33011i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.32e5 + 2.02e5i)T \)
good2 \( 1 + (-12.1 - 6.99i)T + (64 + 110. i)T^{2} \)
3 \( 1 + (-43.2 - 74.9i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (221. - 127. i)T + (3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (32.7 + 56.6i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 4.26e3T + 1.94e7T^{2} \)
13 \( 1 + (-3.54e3 + 2.04e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-8.92e3 - 5.15e3i)T + (2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-5.49e3 + 3.17e3i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 - 1.08e5iT - 3.40e9T^{2} \)
29 \( 1 + 1.79e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.30e5iT - 2.75e10T^{2} \)
41 \( 1 + (2.11e5 + 3.67e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + 6.92e5iT - 2.71e11T^{2} \)
47 \( 1 - 9.64e5T + 5.06e11T^{2} \)
53 \( 1 + (7.39e4 - 1.28e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-4.76e5 - 2.74e5i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-6.34e4 + 3.66e4i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.07e6 + 1.86e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-3.53e4 - 6.13e4i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 2.79e5T + 1.10e13T^{2} \)
79 \( 1 + (3.45e6 - 1.99e6i)T + (9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-2.19e6 + 3.79e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (3.28e6 + 1.89e6i)T + (2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.70e7iT - 8.07e13T^{2} \)
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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

−15.44069983325834787286623005824, −14.37825598792497408257817562343, −13.62916632531510677287880628682, −11.68324020814555174401987592956, −10.27081270788467586719739968434, −9.068729899073905226857927682961, −7.48950603691894807955099422935, −5.52201426664377029024066087439, −3.99494263055333917433716092314, −3.47273108121455981739905497407, 1.15031774077303998993101635870, 2.74328168898941976096190395002, 4.11862861949238782645262296576, 6.35948777131516562085788756947, 7.896425800367482311734793931553, 8.840796097429797616385930904049, 11.53738121430884711649445461742, 12.27916392491467417100441716125, 13.02023566605481562649576501012, 14.11424468414798961661185825755

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