Properties

Label 2-37-37.10-c5-0-9
Degree $2$
Conductor $37$
Sign $0.722 + 0.691i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.51 + 6.09i)2-s + (−0.728 − 1.26i)3-s + (−8.75 − 15.1i)4-s + (−24.4 − 42.2i)5-s + 10.2·6-s + (−37.5 − 65.1i)7-s − 101.·8-s + (120. − 208. i)9-s + 343.·10-s + 394.·11-s + (−12.7 + 22.0i)12-s + (36.2 + 62.7i)13-s + 529.·14-s + (−35.5 + 61.5i)15-s + (638. − 1.10e3i)16-s + (201. − 348. i)17-s + ⋯
L(s)  = 1  + (−0.621 + 1.07i)2-s + (−0.0467 − 0.0809i)3-s + (−0.273 − 0.473i)4-s + (−0.436 − 0.756i)5-s + 0.116·6-s + (−0.289 − 0.502i)7-s − 0.563·8-s + (0.495 − 0.858i)9-s + 1.08·10-s + 0.983·11-s + (−0.0255 + 0.0442i)12-s + (0.0594 + 0.102i)13-s + 0.721·14-s + (−0.0408 + 0.0706i)15-s + (0.623 − 1.08i)16-s + (0.168 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.697507 - 0.280175i\)
\(L(\frac12)\) \(\approx\) \(0.697507 - 0.280175i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-3.14e3 + 7.71e3i)T \)
good2 \( 1 + (3.51 - 6.09i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (0.728 + 1.26i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (24.4 + 42.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (37.5 + 65.1i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 394.T + 1.61e5T^{2} \)
13 \( 1 + (-36.2 - 62.7i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-201. + 348. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.01e3 + 1.75e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 3.99e3T + 6.43e6T^{2} \)
29 \( 1 + 4.55e3T + 2.05e7T^{2} \)
31 \( 1 - 1.79e3T + 2.86e7T^{2} \)
41 \( 1 + (-2.06e3 - 3.57e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 7.46e3T + 1.47e8T^{2} \)
47 \( 1 + 1.26e4T + 2.29e8T^{2} \)
53 \( 1 + (1.05e4 - 1.83e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.36e3 - 2.37e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-9.95e3 - 1.72e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-8.26e3 - 1.43e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.41e4 - 5.91e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 3.99e3T + 2.07e9T^{2} \)
79 \( 1 + (1.28e4 + 2.21e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-7.69e3 + 1.33e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-3.44e4 + 5.96e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.25e5T + 8.58e9T^{2} \)
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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

−15.58644423625852132158578102700, −14.41043720705076940587586229535, −12.76382457694858730298277455397, −11.71889016916330526194752675508, −9.651542790016317956227985657573, −8.705420114032409763859100164599, −7.32357657659373026052535086051, −6.24333215182343656807667863809, −4.08060263193306587188732491552, −0.53935085263662320069698932427, 1.91952457205518541422878649217, 3.68199844543802467843443233891, 6.21640221716608849870769971370, 8.052737731611797455361485965601, 9.600917235085853462752612586615, 10.58442139153831943871779978053, 11.60174790560669014342232420961, 12.63991090103582170156308101493, 14.38698992172394511122108412298, 15.47855867995650650144702095466

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