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Properties

Label 2-37-37.21-c1-0-1
Degree $2$
Conductor $37$
Sign $0.231 - 0.972i$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.841 + 2.31i)2-s + (2.14 − 0.779i)3-s + (−3.10 − 2.60i)4-s + (−2.84 − 0.500i)5-s + 5.60i·6-s + (0.251 − 1.42i)7-s + (4.36 − 2.51i)8-s + (1.67 − 1.40i)9-s + (3.54 − 6.14i)10-s + (1.43 + 2.48i)11-s + (−8.66 − 3.15i)12-s + (−0.863 + 1.02i)13-s + (3.08 + 1.78i)14-s + (−6.47 + 1.14i)15-s + (0.745 + 4.22i)16-s + (0.409 + 0.487i)17-s + ⋯
L(s)  = 1  + (−0.594 + 1.63i)2-s + (1.23 − 0.449i)3-s + (−1.55 − 1.30i)4-s + (−1.27 − 0.224i)5-s + 2.28i·6-s + (0.0950 − 0.539i)7-s + (1.54 − 0.890i)8-s + (0.558 − 0.468i)9-s + (1.12 − 1.94i)10-s + (0.432 + 0.748i)11-s + (−2.50 − 0.910i)12-s + (−0.239 + 0.285i)13-s + (0.824 + 0.475i)14-s + (−1.67 + 0.294i)15-s + (0.186 + 1.05i)16-s + (0.0992 + 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.231 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515047 + 0.407073i\)
\(L(\frac12)\) \(\approx\) \(0.515047 + 0.407073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-4.38 - 4.21i)T \)
good2 \( 1 + (0.841 - 2.31i)T + (-1.53 - 1.28i)T^{2} \)
3 \( 1 + (-2.14 + 0.779i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (2.84 + 0.500i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.251 + 1.42i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.43 - 2.48i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.863 - 1.02i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.409 - 0.487i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-0.636 - 1.74i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (6.38 + 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.44 + 4.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.28iT - 31T^{2} \)
41 \( 1 + (4.07 + 3.41i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 5.16iT - 43T^{2} \)
47 \( 1 + (5.53 - 9.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.46 + 8.29i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-5.35 + 0.943i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-0.559 + 0.666i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.975 + 5.52i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.480 + 0.174i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 3.16T + 73T^{2} \)
79 \( 1 + (-3.22 - 0.567i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.83 - 3.21i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (0.860 - 0.151i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (11.4 + 6.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−16.46425682905192028090476563264, −15.51990608913006003487821403235, −14.58362165979706031185909461216, −13.85180665798958664938494142632, −12.15182590990421579860157330151, −9.825874390548534431515993580238, −8.387921644080676364064047430893, −7.87245643330148126168482106347, −6.79596319845813941279063758426, −4.30588116630336647595851982164, 2.90585490194562999260867628957, 3.96117605352766742286988566884, 8.011075825309524472780970170857, 8.784797970141332334546817190012, 9.952848858741226680606297007987, 11.36820405805270742729210686279, 12.12510212001073026718347764705, 13.67520784095224996506847614552, 14.93238513626759305798210224800, 16.05774833134528136371328122720

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