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Properties

Label 1-269-269.51-r0-0-0
Degree $1$
Conductor $269$
Sign $0.847 - 0.530i$
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  + (0.300 − 0.953i)2-s + (0.972 + 0.232i)3-s + (−0.819 − 0.572i)4-s + (0.845 + 0.533i)5-s + (0.513 − 0.858i)6-s + (0.472 + 0.881i)7-s + (−0.792 + 0.610i)8-s + (0.892 + 0.451i)9-s + (0.762 − 0.646i)10-s + (−0.553 − 0.833i)11-s + (−0.664 − 0.747i)12-s + (−0.762 + 0.646i)13-s + (0.982 − 0.186i)14-s + (0.698 + 0.715i)15-s + (0.344 + 0.938i)16-s + (0.553 − 0.833i)17-s + ⋯
L(s)  = 1  + (0.300 − 0.953i)2-s + (0.972 + 0.232i)3-s + (−0.819 − 0.572i)4-s + (0.845 + 0.533i)5-s + (0.513 − 0.858i)6-s + (0.472 + 0.881i)7-s + (−0.792 + 0.610i)8-s + (0.892 + 0.451i)9-s + (0.762 − 0.646i)10-s + (−0.553 − 0.833i)11-s + (−0.664 − 0.747i)12-s + (−0.762 + 0.646i)13-s + (0.982 − 0.186i)14-s + (0.698 + 0.715i)15-s + (0.344 + 0.938i)16-s + (0.553 − 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.847 - 0.530i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.847 - 0.530i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.983027836 - 0.5698697280i\)
\(L(\frac12)\) \(\approx\) \(1.983027836 - 0.5698697280i\)
\(L(1)\) \(\approx\) \(1.632232028 - 0.4458896832i\)
\(L(1)\) \(\approx\) \(1.632232028 - 0.4458896832i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (0.300 - 0.953i)T \)
3 \( 1 + (0.972 + 0.232i)T \)
5 \( 1 + (0.845 + 0.533i)T \)
7 \( 1 + (0.472 + 0.881i)T \)
11 \( 1 + (-0.553 - 0.833i)T \)
13 \( 1 + (-0.762 + 0.646i)T \)
17 \( 1 + (0.553 - 0.833i)T \)
19 \( 1 + (0.912 + 0.409i)T \)
23 \( 1 + (-0.972 - 0.232i)T \)
29 \( 1 + (-0.430 + 0.902i)T \)
31 \( 1 + (0.388 - 0.921i)T \)
37 \( 1 + (-0.209 - 0.977i)T \)
41 \( 1 + (-0.300 - 0.953i)T \)
43 \( 1 + (0.430 - 0.902i)T \)
47 \( 1 + (-0.946 - 0.322i)T \)
53 \( 1 + (-0.998 - 0.0468i)T \)
59 \( 1 + (0.819 + 0.572i)T \)
61 \( 1 + (-0.869 - 0.493i)T \)
67 \( 1 + (-0.819 + 0.572i)T \)
71 \( 1 + (-0.982 - 0.186i)T \)
73 \( 1 + (0.930 - 0.366i)T \)
79 \( 1 + (0.960 - 0.277i)T \)
83 \( 1 + (-0.591 - 0.806i)T \)
89 \( 1 + (0.0702 - 0.997i)T \)
97 \( 1 + (-0.869 + 0.493i)T \)
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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

−25.80506512388070725215033407649, −24.889524576002351870532280166855, −24.238647575115563035497508381038, −23.492306692015279875669210745616, −22.252034201354411879218968003841, −21.1941934648649261552865278540, −20.53178594621734939817020413, −19.57446174339617115648484427745, −18.00042192652638632721239380402, −17.65381841500553759971863430341, −16.58770266059937390059399851710, −15.43460713916999561548651077078, −14.57812252291886622969298381854, −13.799181181775991177049171786198, −13.08550488793662437629492233870, −12.27670272960367465953817901686, −10.053172551586873700360444461732, −9.62596526783379603657351829020, −8.12416330040067951772495561469, −7.73796131493131844294189851305, −6.56102172062576049398598089272, −5.188609211825938204812412730913, −4.35196950281467883088229989606, −2.964168071046903811004513110210, −1.4401157635546919059675449928, 1.801903531444037707061142639481, 2.55644289442653421219470236248, 3.44796796367567417041295400887, 4.95967189524655384407635084229, 5.794463372722888753285849218909, 7.55432078902505653822739505197, 8.81444671282634704393162349479, 9.55451610684322896383520909204, 10.36186516735103443803689122180, 11.51248969040187540853238219022, 12.54180871316209280095134928507, 13.82164738389261435286294783901, 14.142413442963953056949872227939, 15.02136109469576393302150385020, 16.2772976089229946006266230493, 17.92568071777364373041576170537, 18.610830269393209880789885470550, 19.17525897695878182859461762996, 20.52683836838997766314912458683, 21.06650693198344177801207987948, 21.885281475947592457405808332155, 22.38218545021702960861861100309, 24.08163023924403826726271951897, 24.65835139050537514707054909738, 25.84410932514193401136920616198

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