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Properties

Label 2-37-37.11-c3-0-7
Degree $2$
Conductor $37$
Sign $0.482 + 0.875i$
Analytic cond. $2.18307$
Root an. cond. $1.47752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.00 − 1.15i)2-s + (3.80 − 6.59i)3-s + (−1.33 + 2.30i)4-s + (−1.68 − 0.971i)5-s − 17.5i·6-s + (−1.95 + 3.38i)7-s + 24.6i·8-s + (−15.4 − 26.7i)9-s − 4.48·10-s + 30.7·11-s + (10.1 + 17.5i)12-s + (20.1 + 11.6i)13-s + 9.02i·14-s + (−12.8 + 7.39i)15-s + (17.7 + 30.8i)16-s + (−73.7 + 42.6i)17-s + ⋯
L(s)  = 1  + (0.707 − 0.408i)2-s + (0.732 − 1.26i)3-s + (−0.166 + 0.288i)4-s + (−0.150 − 0.0868i)5-s − 1.19i·6-s + (−0.105 + 0.182i)7-s + 1.08i·8-s + (−0.572 − 0.991i)9-s − 0.141·10-s + 0.843·11-s + (0.243 + 0.422i)12-s + (0.429 + 0.248i)13-s + 0.172i·14-s + (−0.220 + 0.127i)15-s + (0.278 + 0.481i)16-s + (−1.05 + 0.607i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.482 + 0.875i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ 0.482 + 0.875i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.67573 - 0.989546i\)
\(L(\frac12)\) \(\approx\) \(1.67573 - 0.989546i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-167. + 150. i)T \)
good2 \( 1 + (-2.00 + 1.15i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-3.80 + 6.59i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (1.68 + 0.971i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (1.95 - 3.38i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 - 30.7T + 1.33e3T^{2} \)
13 \( 1 + (-20.1 - 11.6i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (73.7 - 42.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (69.7 + 40.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 133. iT - 1.21e4T^{2} \)
29 \( 1 + 287. iT - 2.43e4T^{2} \)
31 \( 1 - 102. iT - 2.97e4T^{2} \)
41 \( 1 + (91.5 - 158. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 349. iT - 7.95e4T^{2} \)
47 \( 1 + 201.T + 1.03e5T^{2} \)
53 \( 1 + (75.3 + 130. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (589. - 340. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-652. - 376. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-252. + 437. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-122. + 211. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 784.T + 3.89e5T^{2} \)
79 \( 1 + (-374. - 215. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-260. - 450. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (944. - 545. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 115. iT - 9.12e5T^{2} \)
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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

−15.28486541977144234001203788253, −13.96951506977398193860178111566, −13.33963040131921708029379815092, −12.38159404640388670085489494958, −11.37834567967295902809722989636, −8.968402170171855971259735578792, −7.977591182552549665909952146928, −6.40486374612985351707979553289, −4.02731996848355388200574396897, −2.19618387465327446304451797709, 3.65844069921818046486523653531, 4.73836541257029714846515285505, 6.55843994450521958097018231782, 8.725252820017126841442415455923, 9.756325514666798354725893174138, 10.95371499836245319676766606213, 12.92413980486354198748782800994, 14.17927477078331682989011001003, 14.85633567425878523882923173971, 15.71608028729149151618918142287

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