Properties

Label 2-37-37.7-c7-0-2
Degree $2$
Conductor $37$
Sign $-0.641 - 0.767i$
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  + (10.2 − 3.73i)2-s + (−19.4 − 7.07i)3-s + (−6.79 + 5.69i)4-s + (7.78 + 44.1i)5-s − 225.·6-s + (−27.0 − 153. i)7-s + (−746. + 1.29e3i)8-s + (−1.34e3 − 1.13e3i)9-s + (244. + 423. i)10-s + (−2.70e3 + 4.68e3i)11-s + (172. − 62.7i)12-s + (−8.54e3 + 7.16e3i)13-s + (−849. − 1.47e3i)14-s + (161. − 913. i)15-s + (−2.63e3 + 1.49e4i)16-s + (6.72e3 + 5.64e3i)17-s + ⋯
L(s)  = 1  + (0.906 − 0.329i)2-s + (−0.415 − 0.151i)3-s + (−0.0530 + 0.0445i)4-s + (0.0278 + 0.157i)5-s − 0.426·6-s + (−0.0297 − 0.168i)7-s + (−0.515 + 0.893i)8-s + (−0.615 − 0.516i)9-s + (0.0773 + 0.133i)10-s + (−0.612 + 1.06i)11-s + (0.0288 − 0.0104i)12-s + (−1.07 + 0.905i)13-s + (−0.0827 − 0.143i)14-s + (0.0123 − 0.0698i)15-s + (−0.160 + 0.911i)16-s + (0.332 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.641 - 0.767i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.641 - 0.767i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.307064 + 0.657144i\)
\(L(\frac12)\) \(\approx\) \(0.307064 + 0.657144i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.75e5 + 1.38e5i)T \)
good2 \( 1 + (-10.2 + 3.73i)T + (98.0 - 82.2i)T^{2} \)
3 \( 1 + (19.4 + 7.07i)T + (1.67e3 + 1.40e3i)T^{2} \)
5 \( 1 + (-7.78 - 44.1i)T + (-7.34e4 + 2.67e4i)T^{2} \)
7 \( 1 + (27.0 + 153. i)T + (-7.73e5 + 2.81e5i)T^{2} \)
11 \( 1 + (2.70e3 - 4.68e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (8.54e3 - 7.16e3i)T + (1.08e7 - 6.17e7i)T^{2} \)
17 \( 1 + (-6.72e3 - 5.64e3i)T + (7.12e7 + 4.04e8i)T^{2} \)
19 \( 1 + (-1.81e4 - 6.59e3i)T + (6.84e8 + 5.74e8i)T^{2} \)
23 \( 1 + (5.12e4 + 8.88e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (2.57e4 - 4.45e4i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + 2.69e5T + 2.75e10T^{2} \)
41 \( 1 + (-3.40e5 + 2.85e5i)T + (3.38e10 - 1.91e11i)T^{2} \)
43 \( 1 - 6.68e5T + 2.71e11T^{2} \)
47 \( 1 + (-7.03e4 - 1.21e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (2.04e5 - 1.16e6i)T + (-1.10e12 - 4.01e11i)T^{2} \)
59 \( 1 + (2.58e4 - 1.46e5i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (2.49e6 - 2.09e6i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (-7.30e5 - 4.14e6i)T + (-5.69e12 + 2.07e12i)T^{2} \)
71 \( 1 + (-6.26e5 - 2.28e5i)T + (6.96e12 + 5.84e12i)T^{2} \)
73 \( 1 + 2.82e6T + 1.10e13T^{2} \)
79 \( 1 + (1.58e5 + 8.98e5i)T + (-1.80e13 + 6.56e12i)T^{2} \)
83 \( 1 + (-3.49e6 - 2.93e6i)T + (4.71e12 + 2.67e13i)T^{2} \)
89 \( 1 + (5.55e5 - 3.14e6i)T + (-4.15e13 - 1.51e13i)T^{2} \)
97 \( 1 + (5.58e6 + 9.67e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−14.73170821480733447149628976385, −14.20253437005177850110426770234, −12.49624487530955653073429013735, −12.19073061322145446122368570067, −10.65672620844976884657816502186, −9.048037965120990291653571622116, −7.24573460078946547024012101183, −5.58089917901167863768857305404, −4.25803013982796122720483416564, −2.49249143542937376090671033286, 0.24003380251878214611019107182, 3.16364027063652481195755667281, 5.16682528492978651543953731369, 5.70330087887418105116994972751, 7.69497730903530065960304911220, 9.432981978616008696973137480496, 10.86623081778802578469262712973, 12.23581647110236717551394355026, 13.40322476332043188377290320431, 14.30122776939394932277615052318

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