Properties

Label 1-37-37.14-r1-0-0
Degree $1$
Conductor $37$
Sign $0.320 - 0.947i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s i·6-s + (−0.5 + 0.866i)7-s i·8-s + (−0.5 − 0.866i)9-s + 10-s − 11-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + i·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s i·6-s + (−0.5 + 0.866i)7-s i·8-s + (−0.5 − 0.866i)9-s + 10-s − 11-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + i·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ 0.320 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.214878642 - 1.588705544i\)
\(L(\frac12)\) \(\approx\) \(2.214878642 - 1.588705544i\)
\(L(1)\) \(\approx\) \(1.788899329 - 0.8842390261i\)
\(L(1)\) \(\approx\) \(1.788899329 - 0.8842390261i\)

Euler product

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

−35.5053932190607241685130514788, −33.65852692686942367308486846130, −33.109954340164283052452655188053, −32.21791630223314197995884199460, −31.16412993949204206322847061738, −29.72826054011695556284856289814, −28.4791402592070744765104733059, −26.5505104733957572481513935030, −25.85376882896272511826575381307, −24.71066482671482379292765836842, −23.206279951262925653279670161448, −22.0211830422638499961370973998, −20.838668062863841128854470877873, −20.17709483797286188836548309907, −17.61291363123304929658793610378, −16.30408133763966006073614798407, −15.47319045165383180818238903034, −13.63825153032010225657527703777, −13.33432963263049683323164711543, −10.95509100079841055132150098543, −9.45916157599402741819201088866, −7.79426069992086584105623131643, −5.82306433277190544103062121737, −4.484115760525524703752214971256, −2.879385196491709182319651624507, 1.937992876899654787260678737377, 3.10701424287336882713489175299, 5.6824733784314758163677025145, 6.7475769032492476719774631092, 8.99777412516335396985127923721, 10.658998884502228069219293882543, 12.35328010993638235746327216750, 13.342609028580464839831913035940, 14.34504630906104646115409071771, 15.735376192637147899707843847643, 18.20823105198513430425428023662, 18.84607377844038018197071116142, 20.39839012866768381106677004500, 21.53658318833374052599655366013, 22.74825255930772399367984767452, 24.07386467213252080654015074150, 25.185994030227197422839716216557, 26.15064172844240294351249134154, 28.77952340206667729225846412995, 28.981540388896056777444053422547, 30.56541542877876940899920486769, 31.169153920152052959432434091802, 32.44782608430526579586468002600, 33.70418417569581147698496796401, 35.01186838611360584385674063343

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