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Properties

Label 1-37-37.19-r1-0-0
Degree $1$
Conductor $37$
Sign $0.512 + 0.858i$
Analytic cond. $3.97620$
Root an. cond. $3.97620$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.173i)2-s + (−0.173 + 0.984i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + i·6-s + (0.766 + 0.642i)7-s + (0.866 − 0.5i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.342 − 0.939i)13-s + (0.866 + 0.5i)14-s + (−0.642 − 0.766i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (−0.173 + 0.984i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + i·6-s + (0.766 + 0.642i)7-s + (0.866 − 0.5i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.342 − 0.939i)13-s + (0.866 + 0.5i)14-s + (−0.642 − 0.766i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ 0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.029642993 + 1.152418744i\)
\(L(\frac12)\) \(\approx\) \(2.029642993 + 1.152418744i\)
\(L(1)\) \(\approx\) \(1.653255768 + 0.5516078989i\)
\(L(1)\) \(\approx\) \(1.653255768 + 0.5516078989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.642 + 0.766i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−34.98346255913779239230214836787, −33.975881811278088004903177203154, −32.56971375497625436208076869893, −31.387207171522633674608902175, −30.49289700059897324324187130127, −29.48118741899830929591542250185, −28.21880505494170342866723872618, −26.44954810403971653616269536635, −24.672714586227949795279618810907, −24.07364326019676342136346819837, −23.327743534559870533724622068408, −21.7146331793357576380902012780, −20.26166330137894580095685872659, −19.29617169491723829261100081393, −17.2562987088248078655300158787, −16.31796750974118115881309770767, −14.43271098264807754216196488123, −13.472970761409397868514957034955, −12.05835175025640528013488945652, −11.30886918557010907795247944203, −8.28910429531704823716689154774, −7.17943611923155048660068156458, −5.51917955363491889972675771657, −3.904718325418339776650738173407, −1.464309442509693827030654868096, 2.81510137788842592327727230104, 4.320472306279818767382239396817, 5.6387110901833294827214847638, 7.57307444164045909047947241601, 9.893761430538187515638509401235, 11.27300951889973917535260613660, 12.109676195511476677689243333766, 14.42247372080694615253223712934, 15.02499065131812529740285527919, 16.130916600865034769333087348662, 18.041822795539283489093770036115, 19.93159114567857326318758135216, 20.86155622401941499786307783068, 22.4337943871783468752265321123, 22.61350622655063081163567690176, 24.355143792537678502559069791500, 25.732087226723672724700472327219, 27.32681985525169004569146119512, 28.14411509315394186251331782393, 29.81784456905107429410710071583, 30.97933580011500989973550983735, 31.79727811854931525651258146105, 33.16197773794718088470250545720, 34.0319207799969995414873483877, 34.90401234442120974284536167365

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