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Properties

Label 1-265-265.193-r0-0-0
Degree $1$
Conductor $265$
Sign $0.0685 - 0.997i$
Analytic cond. $1.23065$
Root an. cond. $1.23065$
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  + (0.970 − 0.239i)2-s + (0.568 − 0.822i)3-s + (0.885 − 0.464i)4-s + (0.354 − 0.935i)6-s + (−0.239 − 0.970i)7-s + (0.748 − 0.663i)8-s + (−0.354 − 0.935i)9-s + (−0.120 + 0.992i)11-s + (0.120 − 0.992i)12-s + (−0.464 + 0.885i)13-s + (−0.464 − 0.885i)14-s + (0.568 − 0.822i)16-s + (0.663 − 0.748i)17-s + (−0.568 − 0.822i)18-s + (−0.464 + 0.885i)19-s + ⋯
L(s)  = 1  + (0.970 − 0.239i)2-s + (0.568 − 0.822i)3-s + (0.885 − 0.464i)4-s + (0.354 − 0.935i)6-s + (−0.239 − 0.970i)7-s + (0.748 − 0.663i)8-s + (−0.354 − 0.935i)9-s + (−0.120 + 0.992i)11-s + (0.120 − 0.992i)12-s + (−0.464 + 0.885i)13-s + (−0.464 − 0.885i)14-s + (0.568 − 0.822i)16-s + (0.663 − 0.748i)17-s + (−0.568 − 0.822i)18-s + (−0.464 + 0.885i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0685 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0685 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $0.0685 - 0.997i$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{265} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ 0.0685 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.835427685 - 1.713716960i\)
\(L(\frac12)\) \(\approx\) \(1.835427685 - 1.713716960i\)
\(L(1)\) \(\approx\) \(1.798927766 - 0.9722135009i\)
\(L(1)\) \(\approx\) \(1.798927766 - 0.9722135009i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.970 - 0.239i)T \)
3 \( 1 + (0.568 - 0.822i)T \)
7 \( 1 + (-0.239 - 0.970i)T \)
11 \( 1 + (-0.120 + 0.992i)T \)
13 \( 1 + (-0.464 + 0.885i)T \)
17 \( 1 + (0.663 - 0.748i)T \)
19 \( 1 + (-0.464 + 0.885i)T \)
23 \( 1 - T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (0.992 - 0.120i)T \)
37 \( 1 + (0.822 + 0.568i)T \)
41 \( 1 + (-0.992 - 0.120i)T \)
43 \( 1 + (0.822 - 0.568i)T \)
47 \( 1 + (0.935 + 0.354i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (-0.663 - 0.748i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (-0.822 + 0.568i)T \)
73 \( 1 + (-0.748 - 0.663i)T \)
79 \( 1 + (0.239 - 0.970i)T \)
83 \( 1 + T \)
89 \( 1 + (0.748 + 0.663i)T \)
97 \( 1 + (-0.935 + 0.354i)T \)
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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

−25.88233779155438013408638061537, −25.056711867858278108397776229006, −24.36864878354278175092913472641, −23.18114652900821116306461162176, −22.03374466164684616372214381250, −21.73846422346851945482272573662, −20.87687309349496087803414566009, −19.781707959998124449386379753774, −19.06501384913764594102180960484, −17.41864472249733929072038374794, −16.36548141428860959499481133634, −15.53407559368177531448917593133, −14.99199804977625245352141368139, −13.98432906313005933457954268585, −13.06381468552550728263306206030, −12.01709941630284208320345714031, −10.93819152065243813834409731881, −9.9203249224271030993793653320, −8.5426096453532444639250257074, −7.86683273135892122142709540852, −6.127448706134144712210776033008, −5.44035580367878661075840188677, −4.2576077440776293508095495230, −3.11684311851250631592825429185, −2.3957752798330195128292728561, 1.373133932124359379172418684832, 2.45091188445519595797516052256, 3.68591643793776353133746355827, 4.6575487206352413623068459729, 6.20761266890972904952845820161, 7.094841080700342644307376969727, 7.78974951031230111983450213926, 9.54145407219057167529146175985, 10.40993582543782579549949936998, 11.928296674810899021115584306698, 12.39303042372102825952128952223, 13.54671594353910761432218533937, 14.14632363766671239792124093117, 14.91412381819464595150000241041, 16.209589069301718253946170102, 17.20187145699271076017573611188, 18.545631869309398919140519987553, 19.39060959739197300267365237289, 20.327113595194601375195264728491, 20.72588226534518314320334491425, 22.063772075042189796005063126430, 23.21427817715292809858098980361, 23.54176376550158974577397489728, 24.54225578687076836333046298345, 25.4810128585024429518898060817

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