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Properties

Label 1-709-709.141-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.622 - 0.782i$
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  + (0.954 + 0.297i)2-s + (−0.0620 + 0.998i)3-s + (0.823 + 0.567i)4-s + (−0.973 + 0.228i)5-s + (−0.355 + 0.934i)6-s + (−0.999 + 0.0177i)7-s + (0.617 + 0.786i)8-s + (−0.992 − 0.123i)9-s + (−0.997 − 0.0709i)10-s + (0.453 + 0.891i)11-s + (−0.617 + 0.786i)12-s + (−0.484 − 0.874i)13-s + (−0.959 − 0.280i)14-s + (−0.167 − 0.985i)15-s + (0.355 + 0.934i)16-s + (−0.545 − 0.838i)17-s + ⋯
L(s)  = 1  + (0.954 + 0.297i)2-s + (−0.0620 + 0.998i)3-s + (0.823 + 0.567i)4-s + (−0.973 + 0.228i)5-s + (−0.355 + 0.934i)6-s + (−0.999 + 0.0177i)7-s + (0.617 + 0.786i)8-s + (−0.992 − 0.123i)9-s + (−0.997 − 0.0709i)10-s + (0.453 + 0.891i)11-s + (−0.617 + 0.786i)12-s + (−0.484 − 0.874i)13-s + (−0.959 − 0.280i)14-s + (−0.167 − 0.985i)15-s + (0.355 + 0.934i)16-s + (−0.545 − 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.622 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2956062905 + 0.6131933972i\)
\(L(\frac12)\) \(\approx\) \(-0.2956062905 + 0.6131933972i\)
\(L(1)\) \(\approx\) \(0.8178181469 + 0.6869676974i\)
\(L(1)\) \(\approx\) \(0.8178181469 + 0.6869676974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.954 + 0.297i)T \)
3 \( 1 + (-0.0620 + 0.998i)T \)
5 \( 1 + (-0.973 + 0.228i)T \)
7 \( 1 + (-0.999 + 0.0177i)T \)
11 \( 1 + (0.453 + 0.891i)T \)
13 \( 1 + (-0.484 - 0.874i)T \)
17 \( 1 + (-0.545 - 0.838i)T \)
19 \( 1 + (-0.372 + 0.928i)T \)
23 \( 1 + (-0.969 - 0.245i)T \)
29 \( 1 + (-0.0974 + 0.995i)T \)
31 \( 1 + (-0.937 - 0.347i)T \)
37 \( 1 + (0.999 - 0.0177i)T \)
41 \( 1 + (-0.322 - 0.946i)T \)
43 \( 1 + (-0.917 - 0.396i)T \)
47 \( 1 + (-0.697 + 0.716i)T \)
53 \( 1 + (0.769 - 0.638i)T \)
59 \( 1 + (-0.617 - 0.786i)T \)
61 \( 1 + (0.405 + 0.914i)T \)
67 \( 1 + (-0.870 + 0.492i)T \)
71 \( 1 + (-0.997 + 0.0709i)T \)
73 \( 1 + (0.0266 + 0.999i)T \)
79 \( 1 + (-0.984 - 0.176i)T \)
83 \( 1 + (0.530 - 0.847i)T \)
89 \( 1 + (0.746 + 0.665i)T \)
97 \( 1 + (-0.150 + 0.988i)T \)
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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

−22.06018618828654777791095944891, −21.65144744232053575806229425256, −20.08046889961674168098055438922, −19.62221628378004866011521638643, −19.2889917574145983825943819522, −18.36439839968199638792665660586, −16.81517410996020652107021822678, −16.4217144648146149873676573809, −15.33487139623781550890079971522, −14.558903572909698807282395299521, −13.45688269465349822675747854536, −13.09075251972743679675078573129, −12.07693430480638308638513737621, −11.62037810182409257843225930098, −10.80800592375849522021358748369, −9.41037977732164390077021365139, −8.377143628045682188977699204, −7.296024765080818036432122130872, −6.529176298365979246700774133863, −5.93695631069010757609204734631, −4.54943510486006904213573602194, −3.67742396345448303503268390753, −2.778625161042620475357403670621, −1.6367386321240342258365653923, −0.218842311650513922659160634772, 2.41640041715372177828594263198, 3.37605004377143269229676355066, 3.992424796927792884280209378564, 4.82877510695233265750125749536, 5.835574588999839491639704282904, 6.83337420090554315541741309367, 7.64830091953233880731595621792, 8.74553672834561481201369674931, 9.919232621833314691290781274930, 10.63450252099249133324900119685, 11.70715844395475195034366254327, 12.29030771793176067765551067759, 13.14283961150277094522995574469, 14.52898157536682532403360912371, 14.83244048885317698219738692420, 15.7725840346796786391069894809, 16.20767127425388547453966425770, 16.97956110079706743498225067194, 18.13968532813816294007788955820, 19.54473633647923710806820273758, 20.19130094602569845848106432527, 20.520899984018124056561763568320, 22.0893792810798037081118588071, 22.22610043430338191378474135634, 22.95909593535344454561641615069

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