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Properties

Label 1-709-709.30-r1-0-0
Degree $1$
Conductor $709$
Sign $-0.198 - 0.980i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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  + (−0.314 − 0.949i)2-s + (−0.0266 + 0.999i)3-s + (−0.802 + 0.596i)4-s + (0.697 + 0.716i)5-s + (0.957 − 0.288i)6-s + (−0.617 + 0.786i)7-s + (0.818 + 0.574i)8-s + (−0.998 − 0.0532i)9-s + (0.461 − 0.887i)10-s + (−0.910 + 0.413i)11-s + (−0.574 − 0.818i)12-s + (−0.786 + 0.617i)13-s + (0.940 + 0.339i)14-s + (−0.734 + 0.678i)15-s + (0.288 − 0.957i)16-s + (0.413 − 0.910i)17-s + ⋯
L(s)  = 1  + (−0.314 − 0.949i)2-s + (−0.0266 + 0.999i)3-s + (−0.802 + 0.596i)4-s + (0.697 + 0.716i)5-s + (0.957 − 0.288i)6-s + (−0.617 + 0.786i)7-s + (0.818 + 0.574i)8-s + (−0.998 − 0.0532i)9-s + (0.461 − 0.887i)10-s + (−0.910 + 0.413i)11-s + (−0.574 − 0.818i)12-s + (−0.786 + 0.617i)13-s + (0.940 + 0.339i)14-s + (−0.734 + 0.678i)15-s + (0.288 − 0.957i)16-s + (0.413 − 0.910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.198 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01526275017 + 0.01865508602i\)
\(L(\frac12)\) \(\approx\) \(0.01526275017 + 0.01865508602i\)
\(L(1)\) \(\approx\) \(0.6488002090 + 0.1520729139i\)
\(L(1)\) \(\approx\) \(0.6488002090 + 0.1520729139i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.314 - 0.949i)T \)
3 \( 1 + (-0.0266 + 0.999i)T \)
5 \( 1 + (0.697 + 0.716i)T \)
7 \( 1 + (-0.617 + 0.786i)T \)
11 \( 1 + (-0.910 + 0.413i)T \)
13 \( 1 + (-0.786 + 0.617i)T \)
17 \( 1 + (0.413 - 0.910i)T \)
19 \( 1 + (0.574 - 0.818i)T \)
23 \( 1 + (-0.106 + 0.994i)T \)
29 \( 1 + (-0.964 - 0.263i)T \)
31 \( 1 + (0.678 + 0.734i)T \)
37 \( 1 + (-0.786 - 0.617i)T \)
41 \( 1 + (-0.507 + 0.861i)T \)
43 \( 1 + (-0.388 + 0.921i)T \)
47 \( 1 + (-0.994 - 0.106i)T \)
53 \( 1 + (-0.678 - 0.734i)T \)
59 \( 1 + (0.574 - 0.818i)T \)
61 \( 1 + (-0.752 - 0.658i)T \)
67 \( 1 + (-0.437 + 0.899i)T \)
71 \( 1 + (0.461 + 0.887i)T \)
73 \( 1 + (0.211 - 0.977i)T \)
79 \( 1 + (0.364 + 0.931i)T \)
83 \( 1 + (0.971 + 0.237i)T \)
89 \( 1 + (0.552 + 0.833i)T \)
97 \( 1 + (0.159 + 0.987i)T \)
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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

−22.79414743425155700445602218197, −22.25610253210210979383323321647, −20.79300025357017383473753897664, −20.06563373697723330441004441852, −19.11437237707186045813317007595, −18.48472268597849468471958084452, −17.58270848119386887247327357824, −16.84830653850593233655394075807, −16.53077068905883262533022752841, −15.27580942571827296550179739679, −14.222649842558979017990748750615, −13.55876978073982568108007525865, −12.93216170518135733252741428160, −12.27281135341248112286520446393, −10.48935524751486270974484789526, −10.04521618782581524830762980163, −8.86140389855237151473717691231, −8.01176477468809174637891980294, −7.437528491327849854688202040988, −6.31924176125610528148246343370, −5.72241819011185609020168900664, −4.86400813170541829382582609406, −3.36950325324801291192764916989, −1.90265794672826344108080697887, −0.7920278802412745129485919580, 0.008001406370654163178862414542, 1.99863348468166624663639566556, 2.83954426309763922721670390726, 3.382269786505579191827066566021, 4.93619606624859465404939481777, 5.34583510440163436110534016041, 6.80199816374506185418097221774, 7.95805292457277806032887387369, 9.42236171464894159341142932780, 9.472023610359615978141111488141, 10.24150988621470541193296261252, 11.28051794036588602066398685233, 11.86009789464748221004549704116, 13.02322551795293809366660245659, 13.84942845580601532121586750104, 14.71894933805348688459097730667, 15.66263902646141775257847292369, 16.48623961111382853472098040443, 17.557286808843404917015125161150, 18.09255283212737358072396436687, 19.0293593901832057192432528966, 19.742781716996911103257618132398, 20.79189378387302430805027983352, 21.3933745634058615041568332517, 21.96541522173351346213080231364

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