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Properties

Label 1-37-37.6-r1-0-0
Degree $1$
Conductor $37$
Sign $0.763 + 0.646i$
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  − i·2-s − 3-s − 4-s + i·5-s + i·6-s + 7-s + i·8-s + 9-s + 10-s − 11-s + 12-s + i·13-s − i·14-s − i·15-s + 16-s + i·17-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s + i·5-s + i·6-s + 7-s + i·8-s + 9-s + 10-s − 11-s + 12-s + i·13-s − i·14-s − i·15-s + 16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.763 + 0.646i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ 0.763 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7412395992 + 0.2717602033i\)
\(L(\frac12)\) \(\approx\) \(0.7412395992 + 0.2717602033i\)
\(L(1)\) \(\approx\) \(0.7279167277 - 0.06024423023i\)
\(L(1)\) \(\approx\) \(0.7279167277 - 0.06024423023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - iT \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + iT \)
17 \( 1 + iT \)
19 \( 1 + T \)
23 \( 1 + iT \)
29 \( 1 + T \)
31 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + iT \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 - iT \)
71 \( 1 + iT \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + iT \)
97 \( 1 - iT \)
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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

−34.88877637373585496863803885232, −34.08580503751473052718054528432, −33.02334902128787778837942243335, −31.97737086532945481397552339744, −30.565618013227519280535083671370, −28.80444057999501108466474263887, −27.79159146636163699922054840962, −26.94437376469512416231180352300, −25.087815753345699111645577512740, −24.10757471078202921369630659360, −23.36159398125948959264241165815, −21.93490635749659673172741735351, −20.58675241728620076806689173337, −18.28297956348607433757262227092, −17.54819346805107344062027090053, −16.34459084233482959740358822823, −15.361026706308112161924320026520, −13.47247533957604169469695772660, −12.29163697425835362595040914215, −10.48657708887078487088265507824, −8.66050249025699236082624712455, −7.31228850687287547825191343163, −5.367288931553414114462919741812, −4.806537687650580890921692256715, −0.61549686593822271927427467532, 1.891032073729348382035324340261, 4.124712373221212554432271434645, 5.740424920407127445224915583427, 7.79307767942996972293028877074, 10.03631058252056740220159506585, 11.023208353879332346728659468945, 11.90892196243595279830171802447, 13.56770498249987919339825994882, 15.05212675722895903193399358668, 17.11826399161001533740919794276, 18.21529580774691769136889862836, 18.9988636491002231108017287498, 21.05488321184147374785129251114, 21.73293222992005180898211617588, 23.0757774706340658982311724958, 23.89416673663820769839409341499, 26.284933176202628419151752563869, 27.25813373171165244577455867558, 28.39385906756649842145484451930, 29.442690082954308381006983715816, 30.43695965808936739084556542574, 31.42351909459715401128376413813, 33.37121270690049889324078612615, 34.14116519199985661833693787526, 35.48889187906396478588624099561

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