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Properties

Label 1-269-269.61-r0-0-0
Degree $1$
Conductor $269$
Sign $0.967 + 0.253i$
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  + (0.960 + 0.277i)2-s + (0.731 − 0.681i)3-s + (0.845 + 0.533i)4-s + (−0.388 + 0.921i)5-s + (0.892 − 0.451i)6-s + (0.591 − 0.806i)7-s + (0.664 + 0.747i)8-s + (0.0702 − 0.997i)9-s + (−0.628 + 0.777i)10-s + (−0.300 + 0.953i)11-s + (0.982 − 0.186i)12-s + (−0.628 + 0.777i)13-s + (0.792 − 0.610i)14-s + (0.344 + 0.938i)15-s + (0.430 + 0.902i)16-s + (−0.300 − 0.953i)17-s + ⋯
L(s)  = 1  + (0.960 + 0.277i)2-s + (0.731 − 0.681i)3-s + (0.845 + 0.533i)4-s + (−0.388 + 0.921i)5-s + (0.892 − 0.451i)6-s + (0.591 − 0.806i)7-s + (0.664 + 0.747i)8-s + (0.0702 − 0.997i)9-s + (−0.628 + 0.777i)10-s + (−0.300 + 0.953i)11-s + (0.982 − 0.186i)12-s + (−0.628 + 0.777i)13-s + (0.792 − 0.610i)14-s + (0.344 + 0.938i)15-s + (0.430 + 0.902i)16-s + (−0.300 − 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.967 + 0.253i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.967 + 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.647396985 + 0.3414481194i\)
\(L(\frac12)\) \(\approx\) \(2.647396985 + 0.3414481194i\)
\(L(1)\) \(\approx\) \(2.138182404 + 0.1836832610i\)
\(L(1)\) \(\approx\) \(2.138182404 + 0.1836832610i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (0.960 + 0.277i)T \)
3 \( 1 + (0.731 - 0.681i)T \)
5 \( 1 + (-0.388 + 0.921i)T \)
7 \( 1 + (0.591 - 0.806i)T \)
11 \( 1 + (-0.300 + 0.953i)T \)
13 \( 1 + (-0.628 + 0.777i)T \)
17 \( 1 + (-0.300 - 0.953i)T \)
19 \( 1 + (0.995 - 0.0936i)T \)
23 \( 1 + (0.731 - 0.681i)T \)
29 \( 1 + (-0.698 + 0.715i)T \)
31 \( 1 + (-0.819 - 0.572i)T \)
37 \( 1 + (-0.998 - 0.0468i)T \)
41 \( 1 + (0.960 - 0.277i)T \)
43 \( 1 + (-0.698 + 0.715i)T \)
47 \( 1 + (-0.912 + 0.409i)T \)
53 \( 1 + (0.163 - 0.986i)T \)
59 \( 1 + (0.845 + 0.533i)T \)
61 \( 1 + (-0.972 - 0.232i)T \)
67 \( 1 + (0.845 - 0.533i)T \)
71 \( 1 + (0.792 + 0.610i)T \)
73 \( 1 + (0.255 - 0.966i)T \)
79 \( 1 + (-0.553 + 0.833i)T \)
83 \( 1 + (-0.990 - 0.140i)T \)
89 \( 1 + (0.513 + 0.858i)T \)
97 \( 1 + (-0.972 + 0.232i)T \)
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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

−25.40211400752493715968235935753, −24.525159144714070996750349240973, −24.23001063916213650540023663983, −22.85586556681996927780213095641, −21.73513172821114012511597859853, −21.33166809512607269194380726861, −20.3964769407876664531983816702, −19.698254031104404550246721206915, −18.82615750294734342175445230264, −17.11875044987622988343742462727, −15.99729933215159996272257637970, −15.39928839714030265161177614228, −14.63231938117850238392422802348, −13.52113213201515564243855929571, −12.7282206279417340270443962109, −11.63957142290244354463208848048, −10.75936234149652303453522880365, −9.49726401728151266961012914086, −8.451657612279988268450579989155, −7.58821684906297326115319137510, −5.50897292195708882063070055670, −5.17467281548103816130500591140, −3.88576842295025006175547682616, −2.95913749817581295153980691414, −1.652306021565458841910337857262, 1.86915652471551631875314949124, 2.866333340158517123644315965750, 3.95973644885326137473476402797, 5.03248303600572782176787815392, 6.93224051498607607303029921081, 7.08105278910481946492255042555, 7.95779064116379877021189151289, 9.58144166310827463954590470335, 11.00043638106038024979403364914, 11.81052025379534058086840869801, 12.85287876600742830889610370768, 13.89155270305072011598240165828, 14.49113498360824534134892996729, 15.109673050323128425234577272388, 16.32502160505512455635115080953, 17.61517042190978995826598158018, 18.43360138204640018838482409303, 19.689174925887719301612181044112, 20.34417030843048367719912861202, 21.15256656394641668382348356065, 22.522333028280949581735225054955, 23.060865475940204733385779299539, 24.09398030705298787288393440747, 24.57049557977251279032389464981, 25.86044673357030874851569032420

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