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Properties

Label 1-137-137.54-r1-0-0
Degree $1$
Conductor $137$
Sign $0.919 + 0.392i$
Analytic cond. $14.7226$
Root an. cond. $14.7226$
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  + (0.961 − 0.273i)2-s + (0.824 + 0.565i)3-s + (0.850 − 0.526i)4-s + (0.486 + 0.873i)5-s + (0.948 + 0.317i)6-s + (0.995 + 0.0922i)7-s + (0.673 − 0.739i)8-s + (0.361 + 0.932i)9-s + (0.707 + 0.707i)10-s + (−0.526 − 0.850i)11-s + (0.998 + 0.0461i)12-s + (−0.769 + 0.638i)13-s + (0.982 − 0.183i)14-s + (−0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + (−0.673 − 0.739i)17-s + ⋯
L(s)  = 1  + (0.961 − 0.273i)2-s + (0.824 + 0.565i)3-s + (0.850 − 0.526i)4-s + (0.486 + 0.873i)5-s + (0.948 + 0.317i)6-s + (0.995 + 0.0922i)7-s + (0.673 − 0.739i)8-s + (0.361 + 0.932i)9-s + (0.707 + 0.707i)10-s + (−0.526 − 0.850i)11-s + (0.998 + 0.0461i)12-s + (−0.769 + 0.638i)13-s + (0.982 − 0.183i)14-s + (−0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + (−0.673 − 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.919 + 0.392i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.919 + 0.392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.718579253 + 0.9637501705i\)
\(L(\frac12)\) \(\approx\) \(4.718579253 + 0.9637501705i\)
\(L(1)\) \(\approx\) \(2.703925168 + 0.2999495373i\)
\(L(1)\) \(\approx\) \(2.703925168 + 0.2999495373i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.961 - 0.273i)T \)
3 \( 1 + (0.824 + 0.565i)T \)
5 \( 1 + (0.486 + 0.873i)T \)
7 \( 1 + (0.995 + 0.0922i)T \)
11 \( 1 + (-0.526 - 0.850i)T \)
13 \( 1 + (-0.769 + 0.638i)T \)
17 \( 1 + (-0.673 - 0.739i)T \)
19 \( 1 + (-0.798 + 0.602i)T \)
23 \( 1 + (0.948 - 0.317i)T \)
29 \( 1 + (-0.317 - 0.948i)T \)
31 \( 1 + (0.990 - 0.138i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.138 + 0.990i)T \)
47 \( 1 + (0.403 - 0.914i)T \)
53 \( 1 + (-0.990 - 0.138i)T \)
59 \( 1 + (0.932 - 0.361i)T \)
61 \( 1 + (0.361 - 0.932i)T \)
67 \( 1 + (-0.638 - 0.769i)T \)
71 \( 1 + (0.228 + 0.973i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (0.565 + 0.824i)T \)
83 \( 1 + (-0.0461 - 0.998i)T \)
89 \( 1 + (-0.873 + 0.486i)T \)
97 \( 1 + (-0.973 - 0.228i)T \)
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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

−28.44443398049945461256761709678, −26.993247144098206687718482567041, −25.64411579088227300430929417038, −25.143432688847657786772662198252, −24.002076020091167078731030329250, −23.78322241427415682178203973292, −22.09196609567868027030438826222, −20.95559066012049378168585470949, −20.49720141076023272490139913151, −19.50985976342545918107653535522, −17.68415727994651836259947258991, −17.19658691593299581568471565466, −15.38023821424427323798101243187, −14.85983079254391049794363357670, −13.619213273255939461207873274923, −12.89956713930573909624419634942, −12.10495666879357010193458283489, −10.485598558765943186937011163, −8.805825472535246037273758153402, −7.8868511522694959925407924399, −6.81949608016945117919600363884, −5.23150801433309809552224108849, −4.38044554519129753371839628354, −2.59110617874205952869183302110, −1.591768988752795205540736853954, 2.05731330341491090559196495386, 2.833033271999269805811313176523, 4.257657295336102328527062981506, 5.31136683059757477278806648677, 6.79271518086646058943623323515, 8.0769576756070489593209852887, 9.62944359837579640823344278814, 10.73806769284430429105487316275, 11.46113243957494001401051064070, 13.205837957685184770669938262329, 14.089602540183282473208801300316, 14.69657618869470055887772006361, 15.56857241525841257891384276938, 16.91806102043874996261712335310, 18.59951450955482179372536892294, 19.3599464157022945522955370114, 20.753701241510825685445501619370, 21.31620597024205746578199507332, 22.00085946891971853027228746469, 23.16510620439809731237377409328, 24.52542347006547463805370708651, 25.03309729843084690607590872177, 26.45912899408633449540488078671, 27.024381669955390787109972055013, 28.50251268807317263697307058263

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