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Properties

Label 1-269-269.11-r0-0-0
Degree $1$
Conductor $269$
Sign $0.883 + 0.469i$
Analytic cond. $1.24923$
Root an. cond. $1.24923$
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  + (0.628 − 0.777i)2-s + (−0.960 − 0.277i)3-s + (−0.209 − 0.977i)4-s + (−0.998 − 0.0468i)5-s + (−0.819 + 0.572i)6-s + (−0.344 + 0.938i)7-s + (−0.892 − 0.451i)8-s + (0.845 + 0.533i)9-s + (−0.664 + 0.747i)10-s + (−0.762 + 0.646i)11-s + (−0.0702 + 0.997i)12-s + (0.664 − 0.747i)13-s + (0.513 + 0.858i)14-s + (0.946 + 0.322i)15-s + (−0.912 + 0.409i)16-s + (0.762 + 0.646i)17-s + ⋯
L(s)  = 1  + (0.628 − 0.777i)2-s + (−0.960 − 0.277i)3-s + (−0.209 − 0.977i)4-s + (−0.998 − 0.0468i)5-s + (−0.819 + 0.572i)6-s + (−0.344 + 0.938i)7-s + (−0.892 − 0.451i)8-s + (0.845 + 0.533i)9-s + (−0.664 + 0.747i)10-s + (−0.762 + 0.646i)11-s + (−0.0702 + 0.997i)12-s + (0.664 − 0.747i)13-s + (0.513 + 0.858i)14-s + (0.946 + 0.322i)15-s + (−0.912 + 0.409i)16-s + (0.762 + 0.646i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.883 + 0.469i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.883 + 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5917353001 + 0.1474411103i\)
\(L(\frac12)\) \(\approx\) \(0.5917353001 + 0.1474411103i\)
\(L(1)\) \(\approx\) \(0.7221616414 - 0.2107511453i\)
\(L(1)\) \(\approx\) \(0.7221616414 - 0.2107511453i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (0.628 - 0.777i)T \)
3 \( 1 + (-0.960 - 0.277i)T \)
5 \( 1 + (-0.998 - 0.0468i)T \)
7 \( 1 + (-0.344 + 0.938i)T \)
11 \( 1 + (-0.762 + 0.646i)T \)
13 \( 1 + (0.664 - 0.747i)T \)
17 \( 1 + (0.762 + 0.646i)T \)
19 \( 1 + (-0.731 + 0.681i)T \)
23 \( 1 + (0.960 + 0.277i)T \)
29 \( 1 + (-0.995 + 0.0936i)T \)
31 \( 1 + (-0.163 + 0.986i)T \)
37 \( 1 + (0.930 + 0.366i)T \)
41 \( 1 + (-0.628 - 0.777i)T \)
43 \( 1 + (0.995 - 0.0936i)T \)
47 \( 1 + (-0.972 + 0.232i)T \)
53 \( 1 + (0.255 + 0.966i)T \)
59 \( 1 + (0.209 + 0.977i)T \)
61 \( 1 + (-0.300 + 0.953i)T \)
67 \( 1 + (-0.209 + 0.977i)T \)
71 \( 1 + (-0.513 + 0.858i)T \)
73 \( 1 + (-0.472 + 0.881i)T \)
79 \( 1 + (-0.0234 - 0.999i)T \)
83 \( 1 + (-0.430 - 0.902i)T \)
89 \( 1 + (-0.388 + 0.921i)T \)
97 \( 1 + (-0.300 - 0.953i)T \)
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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

−25.865997546077106703665601048865, −24.29931589238335879969563298212, −23.6431548251430206574345540079, −23.18519175140036943746619523398, −22.45506602329170241757670130368, −21.28485648006480014873886933829, −20.609795997560216156799203533302, −19.04947275085906735477263904247, −18.21185868368661821656785478399, −16.77047892529764141564496494382, −16.523563379582438951094685809979, −15.668477470998114007180855847986, −14.72943126658358164772430089678, −13.39581337550899251774462675711, −12.72623143539142640896356455997, −11.39699403694976584522940168867, −10.9933251613322062327156815766, −9.390418086612542025304492154834, −7.99846466291791974509281955453, −7.103621502349649841250060295418, −6.27037369103405391857294374590, −5.01256056372821491226486220086, −4.15019893426096218359706931062, −3.26007537682417902578600828902, −0.41435158753084940280834346761, 1.38009791406238039834861281535, 2.8982669117575945131453067896, 4.09448217837860172349418908154, 5.28009235565798133284702637634, 5.96426637047490059151645270676, 7.32896202283565940351280533994, 8.62644325040036949724668370231, 10.14324121603783188017756489617, 10.87988598283608782353528677442, 11.83762202737064455724730573809, 12.66772708389774066216937858708, 12.9915301459416645573978216365, 14.876467144765219210516973536800, 15.445791409718906839602370902825, 16.40297121405867948256993295086, 17.872447295135556974571451275327, 18.758008150093020526820065199395, 19.212181150260120607559722305863, 20.52255356491239583197580462622, 21.36127565345758325289150798202, 22.37569077837817456983521667485, 23.19250732367934181379402062070, 23.46623741579782383954826442607, 24.61221378804259818271896750921, 25.65640076149534603380036386846

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