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Properties

Label 1-767-767.137-r1-0-0
Degree $1$
Conductor $767$
Sign $0.941 + 0.336i$
Analytic cond. $82.4255$
Root an. cond. $82.4255$
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  + (−0.899 − 0.436i)2-s + (−0.891 + 0.452i)3-s + (0.619 + 0.785i)4-s + (0.419 + 0.907i)5-s + (0.999 − 0.0180i)6-s + (0.713 − 0.700i)7-s + (−0.214 − 0.976i)8-s + (0.590 − 0.806i)9-s + (0.0180 − 0.999i)10-s + (0.284 − 0.958i)11-s + (−0.907 − 0.419i)12-s + (−0.947 + 0.319i)14-s + (−0.785 − 0.619i)15-s + (−0.232 + 0.972i)16-s + (0.968 + 0.250i)17-s + (−0.883 + 0.468i)18-s + ⋯
L(s)  = 1  + (−0.899 − 0.436i)2-s + (−0.891 + 0.452i)3-s + (0.619 + 0.785i)4-s + (0.419 + 0.907i)5-s + (0.999 − 0.0180i)6-s + (0.713 − 0.700i)7-s + (−0.214 − 0.976i)8-s + (0.590 − 0.806i)9-s + (0.0180 − 0.999i)10-s + (0.284 − 0.958i)11-s + (−0.907 − 0.419i)12-s + (−0.947 + 0.319i)14-s + (−0.785 − 0.619i)15-s + (−0.232 + 0.972i)16-s + (0.968 + 0.250i)17-s + (−0.883 + 0.468i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 767 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 767 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(767\)    =    \(13 \cdot 59\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(82.4255\)
Root analytic conductor: \(82.4255\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{767} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 767,\ (1:\ ),\ 0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193973836 + 0.2072198350i\)
\(L(\frac12)\) \(\approx\) \(1.193973836 + 0.2072198350i\)
\(L(1)\) \(\approx\) \(0.7069385749 + 0.02476671997i\)
\(L(1)\) \(\approx\) \(0.7069385749 + 0.02476671997i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.899 - 0.436i)T \)
3 \( 1 + (-0.891 + 0.452i)T \)
5 \( 1 + (0.419 + 0.907i)T \)
7 \( 1 + (0.713 - 0.700i)T \)
11 \( 1 + (0.284 - 0.958i)T \)
17 \( 1 + (0.968 + 0.250i)T \)
19 \( 1 + (-0.992 + 0.126i)T \)
23 \( 1 + (0.999 - 0.0361i)T \)
29 \( 1 + (0.997 + 0.0721i)T \)
31 \( 1 + (-0.605 + 0.796i)T \)
37 \( 1 + (-0.953 + 0.302i)T \)
41 \( 1 + (0.847 + 0.530i)T \)
43 \( 1 + (-0.958 + 0.284i)T \)
47 \( 1 + (0.419 - 0.907i)T \)
53 \( 1 + (-0.856 + 0.515i)T \)
61 \( 1 + (0.997 - 0.0721i)T \)
67 \( 1 + (0.738 - 0.674i)T \)
71 \( 1 + (-0.576 - 0.817i)T \)
73 \( 1 + (0.319 + 0.947i)T \)
79 \( 1 + (0.0541 + 0.998i)T \)
83 \( 1 + (0.986 - 0.161i)T \)
89 \( 1 + (0.0721 - 0.997i)T \)
97 \( 1 + (0.661 + 0.750i)T \)
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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

−22.19000023907015233595209406243, −21.09160016044498239597572512345, −20.63809901112704961754601302050, −19.37494167500765522369529104478, −18.79579929712588170387767171356, −17.72720567696348594782415437388, −17.47694114111548733365288574242, −16.73454916913888228009543851936, −15.91003185620500472140242592740, −15.0194927660164674324125327596, −14.117718770986432997865507305439, −12.80137721972659890660580870252, −12.185016359417217164239626223001, −11.389641162552384031584125488821, −10.42698856741131663789841849799, −9.55115593785957087624195948678, −8.700621787282746802717951695781, −7.85725457539322577418989640680, −6.96500932379507966067031239408, −5.99177735971528907278970576143, −5.24608042359907778620987944396, −4.59931801560197589543719914291, −2.265001966707362816551053183924, −1.57898987946530508801944688341, −0.63509987395053239313992135126, 0.744340049721492527480239364560, 1.64045687890742602257843014258, 3.11934670163478455947628908777, 3.860854826040108048931324103514, 5.177211476696026033730898375570, 6.355031635326129741431257953223, 6.92300363632059759866201990723, 8.00902932621412208161701951172, 9.005857434031994523867493718000, 10.11755843493311935177656947277, 10.61836923918215211502110574212, 11.14280152344779918898175982929, 11.93583353002212679096195483453, 12.994387306468271928400282347443, 14.179798741174972157608382834322, 14.98108050562345824649469775989, 16.050577358762631506423997868, 16.98034911239706755391616286245, 17.240034713475171190330351865571, 18.18040310304677030857893107655, 18.82182522532690233571817264745, 19.64455045349610429031427914591, 20.895594271437902023321778452559, 21.38635504861062799713292162175, 21.86881742917406898261244690820

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