Properties

Label 2-37-37.11-c3-0-5
Degree $2$
Conductor $37$
Sign $0.102 + 0.994i$
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  + (−2.74 + 1.58i)2-s + (2.85 − 4.94i)3-s + (1.03 − 1.78i)4-s + (−13.1 − 7.61i)5-s + 18.1i·6-s + (14.3 − 24.8i)7-s − 18.8i·8-s + (−2.78 − 4.81i)9-s + 48.3·10-s + 1.05·11-s + (−5.89 − 10.2i)12-s + (−40.7 − 23.5i)13-s + 91.0i·14-s + (−75.2 + 43.4i)15-s + (38.1 + 66.0i)16-s + (−60.0 + 34.6i)17-s + ⋯
L(s)  = 1  + (−0.971 + 0.560i)2-s + (0.549 − 0.951i)3-s + (0.129 − 0.223i)4-s + (−1.17 − 0.681i)5-s + 1.23i·6-s + (0.775 − 1.34i)7-s − 0.832i·8-s + (−0.103 − 0.178i)9-s + 1.52·10-s + 0.0289·11-s + (−0.141 − 0.245i)12-s + (−0.870 − 0.502i)13-s + 1.73i·14-s + (−1.29 + 0.747i)15-s + (0.595 + 1.03i)16-s + (−0.856 + 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.102 + 0.994i$
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: \((2,\ 37,\ (\ :3/2),\ 0.102 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.519823 - 0.468902i\)
\(L(\frac12)\) \(\approx\) \(0.519823 - 0.468902i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-212. + 73.4i)T \)
good2 \( 1 + (2.74 - 1.58i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-2.85 + 4.94i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (13.1 + 7.61i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-14.3 + 24.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 - 1.05T + 1.33e3T^{2} \)
13 \( 1 + (40.7 + 23.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (60.0 - 34.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-123. - 71.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 123. iT - 1.21e4T^{2} \)
29 \( 1 - 95.4iT - 2.43e4T^{2} \)
31 \( 1 + 87.9iT - 2.97e4T^{2} \)
41 \( 1 + (-112. + 194. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 195. iT - 7.95e4T^{2} \)
47 \( 1 - 293.T + 1.03e5T^{2} \)
53 \( 1 + (-91.3 - 158. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (339. - 196. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-513. - 296. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (181. - 314. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-7.29 + 12.6i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 609.T + 3.89e5T^{2} \)
79 \( 1 + (1.00e3 + 580. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-331. - 573. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-915. + 528. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 976. iT - 9.12e5T^{2} \)
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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

−15.97894892015335436025985389065, −14.50590651103227297297165896705, −13.19621441892030625032599282945, −12.15986033861264666764313315462, −10.44735943839792565191338355886, −8.648530008887309599417762075347, −7.67474243615750884570218653532, −7.33644265534682201524563738815, −4.23188978232392390039694511048, −0.829926286010747690085918799014, 2.78467534314819524517085783887, 4.84190251859872773908032947571, 7.63708101135409044465237107139, 8.938500049317052212342356986703, 9.688379471059461239495366264267, 11.35039492320695646340723831336, 11.70627591485728745410850347535, 14.31583343865347124377093923886, 15.18110499833378120456520641638, 15.83555360435547672546707721596

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