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Properties

Label 1-37-37.35-r1-0-0
Degree $1$
Conductor $37$
Sign $0.0789 - 0.996i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0789 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0789 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.0789 - 0.996i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ 0.0789 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7424946621 - 0.6859944659i\)
\(L(\frac12)\) \(\approx\) \(0.7424946621 - 0.6859944659i\)
\(L(1)\) \(\approx\) \(0.7405587630 - 0.3448346246i\)
\(L(1)\) \(\approx\) \(0.7405587630 - 0.3448346246i\)

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

−35.69265126019711906548978380233, −34.25443625293734794547290623033, −33.3913210225004832575332392516, −32.49433029481102895109984289114, −30.812522218149469970558660232618, −28.93382613784864913166591328158, −28.18311552519937553454207981322, −27.405213041633679852039161724403, −25.94839024926487264403537693880, −25.03807341961331090152119562507, −23.65000435751342850112201361444, −21.52090832484903935324943719480, −20.92961483285619513912190427817, −19.53425270623199901434158644371, −17.66702545048274254585487788417, −17.04086197485321908831969264644, −15.65999750188318630994890376847, −14.54182891544918148362582291803, −12.10267671937071235581071052167, −10.74452865777710369110020323070, −9.31668898970906494136369960411, −8.58851187424847625274515470083, −6.23900896276632979789321686659, −4.70551158111097392973142722457, −1.85723336067379756756939573408, 1.01296372560939616750146823152, 2.72293432365687372094471494947, 6.111995538910197666720272470838, 7.29727657323673763886901764165, 8.6179106495501684897860326766, 10.58344638492487942410959519731, 11.42959955553886523970340098990, 13.26215232727203559158126451401, 14.64494574194757745044474147771, 16.78215966523864947067816517442, 17.764309387085154935933397143505, 18.58088252642171348447316172644, 19.83746308673786457707800612300, 21.21396058961415983459070573831, 22.88228774597401294128126631998, 24.47274016937534036874840622607, 25.29224899683971509032954304917, 26.61125659354295907921302928800, 27.737701693006436982990663555286, 29.4423158124005981118435442874, 29.7645655461738433195241518829, 30.83434067524772617003313019800, 33.12627368920206119726593397671, 34.23629922325635716340407847932, 35.03485256578787262756060956440

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