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Properties

Label 1-709-709.136-r0-0-0
Degree $1$
Conductor $709$
Sign $0.740 - 0.671i$
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  + (0.631 + 0.775i)2-s + (−0.271 + 0.962i)3-s + (−0.202 + 0.979i)4-s + (−0.992 + 0.123i)5-s + (−0.917 + 0.396i)6-s + (−0.560 + 0.828i)7-s + (−0.887 + 0.461i)8-s + (−0.852 − 0.522i)9-s + (−0.722 − 0.691i)10-s + (−0.671 − 0.740i)11-s + (−0.887 − 0.461i)12-s + (−0.437 + 0.899i)13-s + (−0.996 + 0.0886i)14-s + (0.150 − 0.988i)15-s + (−0.917 − 0.396i)16-s + (−0.305 − 0.952i)17-s + ⋯
L(s)  = 1  + (0.631 + 0.775i)2-s + (−0.271 + 0.962i)3-s + (−0.202 + 0.979i)4-s + (−0.992 + 0.123i)5-s + (−0.917 + 0.396i)6-s + (−0.560 + 0.828i)7-s + (−0.887 + 0.461i)8-s + (−0.852 − 0.522i)9-s + (−0.722 − 0.691i)10-s + (−0.671 − 0.740i)11-s + (−0.887 − 0.461i)12-s + (−0.437 + 0.899i)13-s + (−0.996 + 0.0886i)14-s + (0.150 − 0.988i)15-s + (−0.917 − 0.396i)16-s + (−0.305 − 0.952i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1804691334 + 0.06964343539i\)
\(L(\frac12)\) \(\approx\) \(-0.1804691334 + 0.06964343539i\)
\(L(1)\) \(\approx\) \(0.4310587068 + 0.6101261050i\)
\(L(1)\) \(\approx\) \(0.4310587068 + 0.6101261050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.631 + 0.775i)T \)
3 \( 1 + (-0.271 + 0.962i)T \)
5 \( 1 + (-0.992 + 0.123i)T \)
7 \( 1 + (-0.560 + 0.828i)T \)
11 \( 1 + (-0.671 - 0.740i)T \)
13 \( 1 + (-0.437 + 0.899i)T \)
17 \( 1 + (-0.305 - 0.952i)T \)
19 \( 1 + (0.842 + 0.537i)T \)
23 \( 1 + (0.453 + 0.891i)T \)
29 \( 1 + (-0.792 - 0.610i)T \)
31 \( 1 + (0.781 + 0.624i)T \)
37 \( 1 + (-0.560 + 0.828i)T \)
41 \( 1 + (0.710 - 0.703i)T \)
43 \( 1 + (0.895 + 0.445i)T \)
47 \( 1 + (-0.998 - 0.0532i)T \)
53 \( 1 + (-0.931 + 0.364i)T \)
59 \( 1 + (-0.887 + 0.461i)T \)
61 \( 1 + (-0.813 - 0.582i)T \)
67 \( 1 + (0.999 - 0.0354i)T \)
71 \( 1 + (-0.722 + 0.691i)T \)
73 \( 1 + (0.994 + 0.106i)T \)
79 \( 1 + (-0.943 - 0.330i)T \)
83 \( 1 + (-0.617 - 0.786i)T \)
89 \( 1 + (0.684 - 0.728i)T \)
97 \( 1 + (-0.903 - 0.429i)T \)
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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

−22.339932283772110517221500919733, −20.80743488927274112884058905482, −20.06438338686841021337192698100, −19.67396859307585560364795038462, −18.90216116525437995100910970385, −18.01917937809541922914833003161, −17.17242988044375195189760372590, −16.011187192123182280844589472828, −15.15508701740482922023473335291, −14.29866281228406480438288693297, −13.137831106086612053253696736367, −12.79226639751270054235026735301, −12.16788752562979131296517319357, −11.013695712702181709083796595303, −10.6095577462687417959657904579, −9.41049787933412038924799865585, −8.067297882540014183244184860988, −7.32066051232113928777648524973, −6.47068556712561486451166547260, −5.28791222951840056965016070649, −4.45127771915532208878527386760, −3.33223835135608630481700826135, −2.49633794670806349469013692315, −1.1185271144617896912477588232, −0.08648283704062985289483239678, 2.85520840836269892596288303497, 3.33141538897914033337926495902, 4.43447762955141250186528919732, 5.20105283905565844859119260808, 6.01754689373183936458528052882, 7.06134902211523428013879982529, 8.03216235870181289445757681239, 8.97641649389683243522314491903, 9.65348159275975857995098757380, 11.169349278078882736396435807644, 11.71152327420890023750463187203, 12.42409280194026973805889400750, 13.66896404149217884305764239333, 14.45729967276091932479409106790, 15.45724340487279125487646173143, 15.843012239928781993477286267412, 16.322793949685061182079150578689, 17.27940156940473487245190375060, 18.438216517075146087018582873680, 19.168408696759769782707648471824, 20.36249080427582937059595593036, 21.22481850048530946074418584425, 21.8155168741104855727399883031, 22.77126991703085788642988299065, 22.962917893113997313270499901627

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