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Properties

Label 2-37-37.26-c7-0-12
Degree $2$
Conductor $37$
Sign $0.833 - 0.553i$
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  + (3.77 + 6.53i)2-s + (9.73 − 16.8i)3-s + (35.5 − 61.5i)4-s + (−147. + 255. i)5-s + 146.·6-s + (91.4 − 158. i)7-s + 1.50e3·8-s + (904. + 1.56e3i)9-s − 2.22e3·10-s + 5.16e3·11-s + (−691. − 1.19e3i)12-s + (5.31e3 − 9.21e3i)13-s + 1.37e3·14-s + (2.87e3 + 4.97e3i)15-s + (1.11e3 + 1.92e3i)16-s + (1.72e4 + 2.99e4i)17-s + ⋯
L(s)  = 1  + (0.333 + 0.577i)2-s + (0.208 − 0.360i)3-s + (0.277 − 0.481i)4-s + (−0.527 + 0.914i)5-s + 0.277·6-s + (0.100 − 0.174i)7-s + 1.03·8-s + (0.413 + 0.716i)9-s − 0.703·10-s + 1.17·11-s + (−0.115 − 0.200i)12-s + (0.671 − 1.16i)13-s + 0.134·14-s + (0.219 + 0.380i)15-s + (0.0679 + 0.117i)16-s + (0.853 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.553i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.833 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.833 - 0.553i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.833 - 0.553i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.47538 + 0.747056i\)
\(L(\frac12)\) \(\approx\) \(2.47538 + 0.747056i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (6.41e4 + 3.01e5i)T \)
good2 \( 1 + (-3.77 - 6.53i)T + (-64 + 110. i)T^{2} \)
3 \( 1 + (-9.73 + 16.8i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (147. - 255. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-91.4 + 158. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 - 5.16e3T + 1.94e7T^{2} \)
13 \( 1 + (-5.31e3 + 9.21e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-1.72e4 - 2.99e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (1.01e3 - 1.75e3i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + 3.31e4T + 3.40e9T^{2} \)
29 \( 1 + 5.19e4T + 1.72e10T^{2} \)
31 \( 1 + 1.18e5T + 2.75e10T^{2} \)
41 \( 1 + (-2.73e5 + 4.73e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + 7.86e5T + 2.71e11T^{2} \)
47 \( 1 + 8.88e5T + 5.06e11T^{2} \)
53 \( 1 + (-5.68e5 - 9.85e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (3.45e5 + 5.98e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.34e6 + 2.32e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.07e6 + 1.86e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (2.79e6 - 4.84e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + 9.87e5T + 1.10e13T^{2} \)
79 \( 1 + (1.76e6 - 3.05e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (6.38e5 + 1.10e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (4.15e6 + 7.18e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.08e6T + 8.07e13T^{2} \)
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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

−14.84075638416481075439051304454, −14.15042957974728311712318407663, −12.80089178787828786164100489417, −11.09462939676114489892756844092, −10.29065084211595147095303321253, −8.049369658027071064970487629961, −7.05108823017814141146966153604, −5.79679326439147869249985755796, −3.77089074491867095999259598680, −1.54455303778825176432194625320, 1.33271942159794861154093463682, 3.54267855355490372300406077847, 4.53296820063185878733494719634, 6.88746623408973698876968219738, 8.530265906032346941684276943305, 9.643154534354441797750804676556, 11.70152941777950244162381872806, 11.92457968348449590897853601205, 13.34700432496302534764525292111, 14.65542493152326532159694600930

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