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Properties

Label 1-269-269.67-r0-0-0
Degree $1$
Conductor $269$
Sign $0.0121 - 0.999i$
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  + (−0.998 + 0.0468i)2-s + (−0.388 − 0.921i)3-s + (0.995 − 0.0936i)4-s + (−0.946 + 0.322i)5-s + (0.430 + 0.902i)6-s + (−0.628 + 0.777i)7-s + (−0.990 + 0.140i)8-s + (−0.698 + 0.715i)9-s + (0.930 − 0.366i)10-s + (−0.209 + 0.977i)11-s + (−0.472 − 0.881i)12-s + (0.930 − 0.366i)13-s + (0.591 − 0.806i)14-s + (0.664 + 0.747i)15-s + (0.982 − 0.186i)16-s + (−0.209 − 0.977i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0468i)2-s + (−0.388 − 0.921i)3-s + (0.995 − 0.0936i)4-s + (−0.946 + 0.322i)5-s + (0.430 + 0.902i)6-s + (−0.628 + 0.777i)7-s + (−0.990 + 0.140i)8-s + (−0.698 + 0.715i)9-s + (0.930 − 0.366i)10-s + (−0.209 + 0.977i)11-s + (−0.472 − 0.881i)12-s + (0.930 − 0.366i)13-s + (0.591 − 0.806i)14-s + (0.664 + 0.747i)15-s + (0.982 − 0.186i)16-s + (−0.209 − 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.0121 - 0.999i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.0121 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2660318582 - 0.2692823239i\)
\(L(\frac12)\) \(\approx\) \(0.2660318582 - 0.2692823239i\)
\(L(1)\) \(\approx\) \(0.4516769478 - 0.1111272597i\)
\(L(1)\) \(\approx\) \(0.4516769478 - 0.1111272597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.998 + 0.0468i)T \)
3 \( 1 + (-0.388 - 0.921i)T \)
5 \( 1 + (-0.946 + 0.322i)T \)
7 \( 1 + (-0.628 + 0.777i)T \)
11 \( 1 + (-0.209 + 0.977i)T \)
13 \( 1 + (0.930 - 0.366i)T \)
17 \( 1 + (-0.209 - 0.977i)T \)
19 \( 1 + (0.513 - 0.858i)T \)
23 \( 1 + (-0.388 - 0.921i)T \)
29 \( 1 + (0.792 + 0.610i)T \)
31 \( 1 + (-0.912 - 0.409i)T \)
37 \( 1 + (-0.869 - 0.493i)T \)
41 \( 1 + (-0.998 - 0.0468i)T \)
43 \( 1 + (0.792 + 0.610i)T \)
47 \( 1 + (0.0702 - 0.997i)T \)
53 \( 1 + (-0.972 - 0.232i)T \)
59 \( 1 + (0.995 - 0.0936i)T \)
61 \( 1 + (0.845 - 0.533i)T \)
67 \( 1 + (0.995 + 0.0936i)T \)
71 \( 1 + (0.591 + 0.806i)T \)
73 \( 1 + (-0.300 - 0.953i)T \)
79 \( 1 + (0.163 - 0.986i)T \)
83 \( 1 + (-0.0234 - 0.999i)T \)
89 \( 1 + (0.344 - 0.938i)T \)
97 \( 1 + (0.845 + 0.533i)T \)
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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

−26.29049197287988353458145285680, −25.515555018524041782176641486509, −23.91560955058214518985487466306, −23.541753727645670522877021209823, −22.291569664841842741649331560864, −21.14947657203253688118988754340, −20.44332074188987432595877921062, −19.54842295732437059352832228345, −18.81703169450446929192794876042, −17.47525824961093541120135916530, −16.59657980946932135579035688621, −16.0233524737939632147748216733, −15.46135130071314424816123824224, −13.99670226665764097387166087675, −12.482241582485250279620488168652, −11.41789465960254382318627603128, −10.78852817715283444922671138741, −9.88883082358845412340059990821, −8.76487002625044620826026298028, −8.02271557803195576191482844275, −6.66642309413635464153232489205, −5.652504630818704886935108109485, −3.85445822722085203432665035939, −3.40871140933868331404085873444, −1.06781647475174499587663751936, 0.456980105598020849825756014113, 2.16344020879188144888714692975, 3.15984719537469752959988766398, 5.2218093492833866026359687898, 6.56471819770903681639818051577, 7.12422874929127933564400506639, 8.16634066680219924188195733575, 9.056838830592042168307461079850, 10.421630928503233912027174403823, 11.411331427285473280477803391660, 12.11941804334929117332104257323, 12.96947512518578760014997028784, 14.56864696567333721723882000842, 15.7949712492487485285292596752, 16.10358057454411214960494961669, 17.60611830823039233590649120083, 18.32720895861241724966985383233, 18.80445881222443240741026753102, 19.871961066948446625543838276416, 20.414061333730821330315706938178, 22.15884762765744999104766142123, 22.95918295397949485084879010646, 23.83380602493245292235248863184, 24.80501687121258701706235194340, 25.5672656531228667530342647431

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