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Properties

Label 1-37-37.11-r0-0-0
Degree $1$
Conductor $37$
Sign $-0.227 - 0.973i$
Analytic cond. $0.171827$
Root an. cond. $0.171827$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s − 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + 10-s + 11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s − 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + 10-s + 11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ -0.227 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5333935533 - 0.6722919873i\)
\(L(\frac12)\) \(\approx\) \(0.5333935533 - 0.6722919873i\)
\(L(1)\) \(\approx\) \(0.8257527772 - 0.6405158417i\)
\(L(1)\) \(\approx\) \(0.8257527772 - 0.6405158417i\)

Euler product

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

−35.37960647576681373815770500470, −34.762161609077013758516753186528, −33.27198739753806512434563354945, −32.49578262225576563306690291030, −31.86070105448124645326974984195, −30.09135481700403175827622341575, −28.38238056910871311235769675193, −27.62725949029154666720638875629, −25.979330948418371416063778591340, −25.06674099066770339338501292526, −23.77754459583210092370914912622, −22.28621897112257085803502222934, −21.72229149467903678741877988342, −20.29870997420394571246548053788, −17.94173665219687310317871226061, −16.844665268180984527579374825567, −15.901816791233388735263243357884, −14.76847431985910955579064076048, −13.03925704377557327295413530883, −11.87621954680345644582451526234, −9.62025576478237367621362735846, −8.60256959646067875006524630929, −6.162451767519408377130020799056, −5.31770132375379024887528943918, −3.68644886799949925299815425593, 1.69586588371255119827752162497, 3.640243487715708306167916876302, 5.87645866112938118136627656081, 7.0314285424670551395856177591, 9.63512103103057395825200324380, 10.977747323450255683364090026737, 12.079044338176255549538565985252, 13.665358401244675493857287208024, 14.22214598167450602970976576640, 16.697647129459961832126046312989, 18.22063333667198676708554413357, 19.056773431281831046346925484221, 20.35757108295103284964384480464, 22.09406078448922288051096786137, 22.805311146996051490635451853243, 23.88879371986542065857688785144, 25.43094993457252915467398723225, 27.03164939033504741324857272104, 28.54648883341156951566987477998, 29.67780723545046883484629249103, 30.03361930708600356299076650270, 31.3070323837560354588740082740, 32.9806642732936194466928833685, 33.81709401786322632447698068814, 35.57894616940229558340879198095

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