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Properties

Label 1-269-269.190-r0-0-0
Degree $1$
Conductor $269$
Sign $-0.462 + 0.886i$
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  + (−0.762 − 0.646i)2-s + (−0.300 − 0.953i)3-s + (0.163 + 0.986i)4-s + (−0.209 + 0.977i)5-s + (−0.388 + 0.921i)6-s + (−0.698 − 0.715i)7-s + (0.513 − 0.858i)8-s + (−0.819 + 0.572i)9-s + (0.792 − 0.610i)10-s + (−0.0234 − 0.999i)11-s + (0.892 − 0.451i)12-s + (0.792 − 0.610i)13-s + (0.0702 + 0.997i)14-s + (0.995 − 0.0936i)15-s + (−0.946 + 0.322i)16-s + (−0.0234 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.762 − 0.646i)2-s + (−0.300 − 0.953i)3-s + (0.163 + 0.986i)4-s + (−0.209 + 0.977i)5-s + (−0.388 + 0.921i)6-s + (−0.698 − 0.715i)7-s + (0.513 − 0.858i)8-s + (−0.819 + 0.572i)9-s + (0.792 − 0.610i)10-s + (−0.0234 − 0.999i)11-s + (0.892 − 0.451i)12-s + (0.792 − 0.610i)13-s + (0.0702 + 0.997i)14-s + (0.995 − 0.0936i)15-s + (−0.946 + 0.322i)16-s + (−0.0234 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $-0.462 + 0.886i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ -0.462 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02997111175 - 0.04943468689i\)
\(L(\frac12)\) \(\approx\) \(0.02997111175 - 0.04943468689i\)
\(L(1)\) \(\approx\) \(0.3935074777 - 0.2142576014i\)
\(L(1)\) \(\approx\) \(0.3935074777 - 0.2142576014i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.762 - 0.646i)T \)
3 \( 1 + (-0.300 - 0.953i)T \)
5 \( 1 + (-0.209 + 0.977i)T \)
7 \( 1 + (-0.698 - 0.715i)T \)
11 \( 1 + (-0.0234 - 0.999i)T \)
13 \( 1 + (0.792 - 0.610i)T \)
17 \( 1 + (-0.0234 + 0.999i)T \)
19 \( 1 + (-0.972 + 0.232i)T \)
23 \( 1 + (-0.300 - 0.953i)T \)
29 \( 1 + (-0.912 + 0.409i)T \)
31 \( 1 + (-0.998 + 0.0468i)T \)
37 \( 1 + (-0.116 + 0.993i)T \)
41 \( 1 + (-0.762 + 0.646i)T \)
43 \( 1 + (-0.912 + 0.409i)T \)
47 \( 1 + (-0.869 - 0.493i)T \)
53 \( 1 + (0.930 - 0.366i)T \)
59 \( 1 + (0.163 + 0.986i)T \)
61 \( 1 + (-0.553 + 0.833i)T \)
67 \( 1 + (0.163 - 0.986i)T \)
71 \( 1 + (0.0702 - 0.997i)T \)
73 \( 1 + (-0.990 + 0.140i)T \)
79 \( 1 + (-0.628 + 0.777i)T \)
83 \( 1 + (0.344 - 0.938i)T \)
89 \( 1 + (0.845 - 0.533i)T \)
97 \( 1 + (-0.553 - 0.833i)T \)
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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

−26.22277993097132649859859888102, −25.6253665135134721840853804283, −24.84478578518635049200038944706, −23.540633202878354062017652247988, −23.05344368669220131013734010521, −21.81824491139950831831173071604, −20.70160981502936180472343241385, −20.042878644492336677959379851760, −18.97155120429827035279373766689, −17.865443036040563559235780243369, −16.95376934674611637104320626816, −16.13428364940148177479353316638, −15.63307463524371974024167494235, −14.78300119685846210907432906140, −13.34741855962522828626434545589, −12.039173187123367185460548361729, −11.129489042473300934356427135489, −9.755283158324517611297684514614, −9.27036690924450326318642879280, −8.50703727007780474613270595942, −7.05065355705433310105953065901, −5.83821330321373892980626164603, −5.0534680827217438309402801100, −3.90287526505627415578712335646, −1.95726280660215165190793223645, 0.05130442691159820262330107837, 1.59297694797647554448682150952, 2.98859695625477674413515477881, 3.76320489712361256342706334104, 6.12692026289448720872383029218, 6.74828031671461488447990982089, 7.88900947452316389204553087578, 8.61613811672711636803029495911, 10.39197294405016995327720258785, 10.738649463082106129967966044372, 11.72172940679577218062024055298, 12.968301237488666307305567829704, 13.436445004077113054085124063945, 14.83225682046334658978283108171, 16.37912587012908757580164266606, 16.94376994203716266166886943583, 18.18086590399787621809955799749, 18.663111996023090067918301934486, 19.474662511155025486029509384750, 20.155701100798225187672348085363, 21.55883265730895870333035002640, 22.45365892710099650108740714287, 23.211233886413418214997296514870, 24.21453608411423254202670753143, 25.64827525764393829049142469258

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