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Properties

Label 1-269-269.25-r0-0-0
Degree $1$
Conductor $269$
Sign $0.591 + 0.806i$
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  + (−0.946 − 0.322i)2-s + (0.344 + 0.938i)3-s + (0.792 + 0.610i)4-s + (0.664 − 0.747i)5-s + (−0.0234 − 0.999i)6-s + (−0.998 + 0.0468i)7-s + (−0.553 − 0.833i)8-s + (−0.762 + 0.646i)9-s + (−0.869 + 0.493i)10-s + (0.995 + 0.0936i)11-s + (−0.300 + 0.953i)12-s + (−0.869 + 0.493i)13-s + (0.960 + 0.277i)14-s + (0.930 + 0.366i)15-s + (0.255 + 0.966i)16-s + (0.995 − 0.0936i)17-s + ⋯
L(s)  = 1  + (−0.946 − 0.322i)2-s + (0.344 + 0.938i)3-s + (0.792 + 0.610i)4-s + (0.664 − 0.747i)5-s + (−0.0234 − 0.999i)6-s + (−0.998 + 0.0468i)7-s + (−0.553 − 0.833i)8-s + (−0.762 + 0.646i)9-s + (−0.869 + 0.493i)10-s + (0.995 + 0.0936i)11-s + (−0.300 + 0.953i)12-s + (−0.869 + 0.493i)13-s + (0.960 + 0.277i)14-s + (0.930 + 0.366i)15-s + (0.255 + 0.966i)16-s + (0.995 − 0.0936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.591 + 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8015144531 + 0.4062327426i\)
\(L(\frac12)\) \(\approx\) \(0.8015144531 + 0.4062327426i\)
\(L(1)\) \(\approx\) \(0.8060739193 + 0.1684579865i\)
\(L(1)\) \(\approx\) \(0.8060739193 + 0.1684579865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.946 - 0.322i)T \)
3 \( 1 + (0.344 + 0.938i)T \)
5 \( 1 + (0.664 - 0.747i)T \)
7 \( 1 + (-0.998 + 0.0468i)T \)
11 \( 1 + (0.995 + 0.0936i)T \)
13 \( 1 + (-0.869 + 0.493i)T \)
17 \( 1 + (0.995 - 0.0936i)T \)
19 \( 1 + (0.591 + 0.806i)T \)
23 \( 1 + (0.344 + 0.938i)T \)
29 \( 1 + (-0.116 + 0.993i)T \)
31 \( 1 + (0.982 + 0.186i)T \)
37 \( 1 + (0.892 - 0.451i)T \)
41 \( 1 + (-0.946 + 0.322i)T \)
43 \( 1 + (-0.116 + 0.993i)T \)
47 \( 1 + (-0.472 - 0.881i)T \)
53 \( 1 + (0.0702 + 0.997i)T \)
59 \( 1 + (0.792 + 0.610i)T \)
61 \( 1 + (-0.698 - 0.715i)T \)
67 \( 1 + (0.792 - 0.610i)T \)
71 \( 1 + (0.960 - 0.277i)T \)
73 \( 1 + (0.845 + 0.533i)T \)
79 \( 1 + (-0.912 - 0.409i)T \)
83 \( 1 + (0.163 - 0.986i)T \)
89 \( 1 + (-0.628 + 0.777i)T \)
97 \( 1 + (-0.698 + 0.715i)T \)
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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

−25.54983645051050437708796917597, −25.01725865458595424478126346190, −24.225791915668109485987772334890, −22.94517208459633960447777268890, −22.22099932469240456405388849507, −20.71721292705036131840181778972, −19.67548587628354566112163531172, −19.12028216155257466961754330616, −18.38986213912212634200103279819, −17.33222644195782764606331974501, −16.841488807383856575614409435676, −15.31438045944764875472482237507, −14.50510613633408842993590697205, −13.62867720418726667463847752542, −12.38767584579850067080608335903, −11.40609782988872800539021718833, −9.97909860898458092507974755588, −9.51027142167221414255830486472, −8.26191233457106962063455862018, −7.08667339685163257622293638884, −6.5905509990814865524171553967, −5.654224206388873956912577799083, −3.14714843474566056411291146610, −2.369103174253645225379328465075, −0.87857034533153734640102321551, 1.41386644071963523574807464992, 2.85161304375179622843440625116, 3.85757756404065307166548008044, 5.341282159984365493591945464477, 6.572553916341965924413179896709, 7.94005151619540359764233394949, 9.17371828831621859471252797887, 9.57607377285171639852370824631, 10.19495024704064301896427947733, 11.69217395222446069937429772458, 12.47646562574689182075149836906, 13.79427072090814614955860116068, 14.89599953986557981844443615165, 16.19173331807527999638152711457, 16.63807858706935458983843949476, 17.3073450679167855943665826620, 18.727149191703040715756265391246, 19.76826977115364921194050478763, 20.14255201623883984968116732279, 21.40416375656379020451863211043, 21.738837601053006666558620051527, 22.93201425781810005510508402145, 24.67068418580781745415373241598, 25.24299412923943063959244974367, 25.924893295211685604111598576645

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