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Properties

Label 1-709-709.133-r0-0-0
Degree $1$
Conductor $709$
Sign $0.896 + 0.443i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (0.999 − 0.0354i)2-s + (0.984 + 0.176i)3-s + (0.997 − 0.0709i)4-s + (0.421 + 0.906i)5-s + (0.989 + 0.141i)6-s + (−0.271 − 0.962i)7-s + (0.994 − 0.106i)8-s + (0.937 + 0.347i)9-s + (0.453 + 0.891i)10-s + (0.220 + 0.975i)11-s + (0.994 + 0.106i)12-s + (−0.697 − 0.716i)13-s + (−0.305 − 0.952i)14-s + (0.254 + 0.967i)15-s + (0.989 − 0.141i)16-s + (−0.954 + 0.297i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0354i)2-s + (0.984 + 0.176i)3-s + (0.997 − 0.0709i)4-s + (0.421 + 0.906i)5-s + (0.989 + 0.141i)6-s + (−0.271 − 0.962i)7-s + (0.994 − 0.106i)8-s + (0.937 + 0.347i)9-s + (0.453 + 0.891i)10-s + (0.220 + 0.975i)11-s + (0.994 + 0.106i)12-s + (−0.697 − 0.716i)13-s + (−0.305 − 0.952i)14-s + (0.254 + 0.967i)15-s + (0.989 − 0.141i)16-s + (−0.954 + 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.896 + 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.932518461 + 0.9196557450i\)
\(L(\frac12)\) \(\approx\) \(3.932518461 + 0.9196557450i\)
\(L(1)\) \(\approx\) \(2.662164697 + 0.3599087547i\)
\(L(1)\) \(\approx\) \(2.662164697 + 0.3599087547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.999 - 0.0354i)T \)
3 \( 1 + (0.984 + 0.176i)T \)
5 \( 1 + (0.421 + 0.906i)T \)
7 \( 1 + (-0.271 - 0.962i)T \)
11 \( 1 + (0.220 + 0.975i)T \)
13 \( 1 + (-0.697 - 0.716i)T \)
17 \( 1 + (-0.954 + 0.297i)T \)
19 \( 1 + (-0.405 + 0.914i)T \)
23 \( 1 + (0.758 + 0.651i)T \)
29 \( 1 + (-0.746 - 0.665i)T \)
31 \( 1 + (0.710 - 0.703i)T \)
37 \( 1 + (-0.271 - 0.962i)T \)
41 \( 1 + (-0.917 - 0.396i)T \)
43 \( 1 + (0.0443 - 0.999i)T \)
47 \( 1 + (0.185 - 0.982i)T \)
53 \( 1 + (-0.964 - 0.263i)T \)
59 \( 1 + (0.994 - 0.106i)T \)
61 \( 1 + (0.823 + 0.567i)T \)
67 \( 1 + (-0.992 + 0.123i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 + (-0.931 - 0.364i)T \)
79 \( 1 + (0.924 - 0.380i)T \)
83 \( 1 + (-0.0266 + 0.999i)T \)
89 \( 1 + (0.959 + 0.280i)T \)
97 \( 1 + (-0.999 + 0.0177i)T \)
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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

−22.10108892182319015844302852614, −21.818255649588275175403465876155, −21.02290452307844547522307969333, −20.29395487677083376609913105583, −19.42917597475695694354006662921, −18.90782158832258451158900014238, −17.54677930155997183937123096464, −16.48274596246218333983902991253, −15.83502956878173194137163368992, −14.99914384993472203015156097451, −14.215635220370746817008402752115, −13.37570378182317163527288076192, −12.87551728526254066928464787368, −12.05293310227533984903552670574, −11.134393092139764321569043600460, −9.72391137576157971050909644196, −8.88972073364005147226964728188, −8.35086411551589767782287248523, −6.9167550988228004754148515433, −6.30081288129443320248438370943, −5.03485524979336941184603529000, −4.452951158710473237450681665009, −3.093996507389680475442924364573, −2.44717132639683626975196792656, −1.43693117546133887847905273133, 1.79913589568226339285965463021, 2.462338587741256950997983430420, 3.578122966803770478960532322376, 4.10673523716065510641559727246, 5.27522501315057677453707707268, 6.58323797460031982270359461308, 7.20385155368754214071320433524, 7.876993999747063339394646510691, 9.494174975807599029700409092367, 10.24934955789071938139869530036, 10.75511380320034758970932933867, 12.07273041762317167188002554099, 13.20563068204594667942655877298, 13.46489398590818773374094527221, 14.55786389606994087908929175371, 14.9640782965316848882889761822, 15.62618027083658630897212245155, 16.92601070403387874513807623196, 17.623948797110257177837300344116, 19.06163328686961845502892518961, 19.557061213477443852250229862047, 20.487048668035498712410533129084, 20.893705580639880989460476351238, 22.04400520267877104835997021521, 22.52489613964729207696959542937

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