Properties

Label 2-37-37.7-c7-0-1
Degree $2$
Conductor $37$
Sign $0.294 - 0.955i$
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  + (−4.25 + 1.54i)2-s + (−41.1 − 14.9i)3-s + (−82.3 + 69.1i)4-s + (−92.7 − 525. i)5-s + 197.·6-s + (9.66 + 54.8i)7-s + (532. − 922. i)8-s + (−209. − 175. i)9-s + (1.20e3 + 2.09e3i)10-s + (−2.79e3 + 4.84e3i)11-s + (4.42e3 − 1.60e3i)12-s + (−1.37e3 + 1.15e3i)13-s + (−125. − 218. i)14-s + (−4.05e3 + 2.30e4i)15-s + (1.55e3 − 8.81e3i)16-s + (1.16e4 + 9.75e3i)17-s + ⋯
L(s)  = 1  + (−0.375 + 0.136i)2-s + (−0.878 − 0.319i)3-s + (−0.643 + 0.540i)4-s + (−0.331 − 1.88i)5-s + 0.373·6-s + (0.0106 + 0.0604i)7-s + (0.367 − 0.637i)8-s + (−0.0958 − 0.0804i)9-s + (0.381 + 0.661i)10-s + (−0.633 + 1.09i)11-s + (0.738 − 0.268i)12-s + (−0.173 + 0.145i)13-s + (−0.0122 − 0.0212i)14-s + (−0.310 + 1.76i)15-s + (0.0948 − 0.537i)16-s + (0.574 + 0.481i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.294 - 0.955i$
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: \((2,\ 37,\ (\ :7/2),\ 0.294 - 0.955i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.256351 + 0.189307i\)
\(L(\frac12)\) \(\approx\) \(0.256351 + 0.189307i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (3.32e4 + 3.06e5i)T \)
good2 \( 1 + (4.25 - 1.54i)T + (98.0 - 82.2i)T^{2} \)
3 \( 1 + (41.1 + 14.9i)T + (1.67e3 + 1.40e3i)T^{2} \)
5 \( 1 + (92.7 + 525. i)T + (-7.34e4 + 2.67e4i)T^{2} \)
7 \( 1 + (-9.66 - 54.8i)T + (-7.73e5 + 2.81e5i)T^{2} \)
11 \( 1 + (2.79e3 - 4.84e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (1.37e3 - 1.15e3i)T + (1.08e7 - 6.17e7i)T^{2} \)
17 \( 1 + (-1.16e4 - 9.75e3i)T + (7.12e7 + 4.04e8i)T^{2} \)
19 \( 1 + (-4.46e3 - 1.62e3i)T + (6.84e8 + 5.74e8i)T^{2} \)
23 \( 1 + (-2.05e4 - 3.55e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-9.51e4 + 1.64e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 - 9.36e4T + 2.75e10T^{2} \)
41 \( 1 + (-4.09e4 + 3.43e4i)T + (3.38e10 - 1.91e11i)T^{2} \)
43 \( 1 + 8.28e5T + 2.71e11T^{2} \)
47 \( 1 + (-1.71e5 - 2.96e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (1.78e5 - 1.00e6i)T + (-1.10e12 - 4.01e11i)T^{2} \)
59 \( 1 + (4.44e5 - 2.51e6i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (2.91e5 - 2.44e5i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (-2.90e5 - 1.64e6i)T + (-5.69e12 + 2.07e12i)T^{2} \)
71 \( 1 + (-3.23e5 - 1.17e5i)T + (6.96e12 + 5.84e12i)T^{2} \)
73 \( 1 - 3.99e5T + 1.10e13T^{2} \)
79 \( 1 + (-4.37e5 - 2.48e6i)T + (-1.80e13 + 6.56e12i)T^{2} \)
83 \( 1 + (4.43e6 + 3.72e6i)T + (4.71e12 + 2.67e13i)T^{2} \)
89 \( 1 + (2.20e6 - 1.25e7i)T + (-4.15e13 - 1.51e13i)T^{2} \)
97 \( 1 + (-1.94e5 - 3.36e5i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−15.52334560785796632566129215929, −13.41785975167900561456141555943, −12.45893429319409861136734160096, −11.97025199988515550186107826708, −9.819427472381388933957449841358, −8.668890669490585626529694487840, −7.56711855135058435568811423345, −5.43215751709942509029765528457, −4.35270255005629158709027161440, −0.998079742461217317391455221560, 0.23935885280213290660558459748, 3.03413932947797708426114042577, 5.16926087874564501001273208298, 6.48672221290529061385728407974, 8.166023825106859456505902152190, 10.15631431803789640539287126226, 10.71984932705472272594810413422, 11.58563259797546007009463920492, 13.76003173334994553552173648498, 14.54565082062935314757659900295

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