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Properties

Label 1-709-709.39-r1-0-0
Degree $1$
Conductor $709$
Sign $-0.102 + 0.994i$
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  + (0.857 + 0.515i)2-s + (0.421 − 0.906i)3-s + (0.468 + 0.883i)4-s + (−0.0443 − 0.999i)5-s + (0.828 − 0.560i)6-s + (−0.992 + 0.123i)7-s + (−0.0532 + 0.998i)8-s + (−0.645 − 0.764i)9-s + (0.476 − 0.879i)10-s + (−0.150 − 0.988i)11-s + (0.998 − 0.0532i)12-s + (−0.921 + 0.388i)13-s + (−0.914 − 0.405i)14-s + (−0.924 − 0.380i)15-s + (−0.560 + 0.828i)16-s + (−0.624 + 0.781i)17-s + ⋯
L(s)  = 1  + (0.857 + 0.515i)2-s + (0.421 − 0.906i)3-s + (0.468 + 0.883i)4-s + (−0.0443 − 0.999i)5-s + (0.828 − 0.560i)6-s + (−0.992 + 0.123i)7-s + (−0.0532 + 0.998i)8-s + (−0.645 − 0.764i)9-s + (0.476 − 0.879i)10-s + (−0.150 − 0.988i)11-s + (0.998 − 0.0532i)12-s + (−0.921 + 0.388i)13-s + (−0.914 − 0.405i)14-s + (−0.924 − 0.380i)15-s + (−0.560 + 0.828i)16-s + (−0.624 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.102 + 0.994i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.102 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.156412449 + 1.281308461i\)
\(L(\frac12)\) \(\approx\) \(1.156412449 + 1.281308461i\)
\(L(1)\) \(\approx\) \(1.446799788 + 0.06001407405i\)
\(L(1)\) \(\approx\) \(1.446799788 + 0.06001407405i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.857 + 0.515i)T \)
3 \( 1 + (0.421 - 0.906i)T \)
5 \( 1 + (-0.0443 - 0.999i)T \)
7 \( 1 + (-0.992 + 0.123i)T \)
11 \( 1 + (-0.150 - 0.988i)T \)
13 \( 1 + (-0.921 + 0.388i)T \)
17 \( 1 + (-0.624 + 0.781i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (0.985 + 0.167i)T \)
29 \( 1 + (0.631 - 0.775i)T \)
31 \( 1 + (0.610 + 0.792i)T \)
37 \( 1 + (0.123 + 0.992i)T \)
41 \( 1 + (0.665 + 0.746i)T \)
43 \( 1 + (-0.959 + 0.280i)T \)
47 \( 1 + (0.769 + 0.638i)T \)
53 \( 1 + (0.991 - 0.132i)T \)
59 \( 1 + (-0.998 - 0.0532i)T \)
61 \( 1 + (0.975 + 0.220i)T \)
67 \( 1 + (0.895 - 0.445i)T \)
71 \( 1 + (0.476 + 0.879i)T \)
73 \( 1 + (-0.982 + 0.185i)T \)
79 \( 1 + (-0.946 + 0.322i)T \)
83 \( 1 + (-0.716 - 0.697i)T \)
89 \( 1 + (-0.928 - 0.372i)T \)
97 \( 1 + (-0.492 - 0.870i)T \)
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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

−22.19099737962555450323083624436, −21.63736722735272169427089449365, −20.53026533707781414907068949031, −19.87757303345109775602062735154, −19.387741691135382493322836260255, −18.34871576525640714871113721382, −17.22251445757203496768372870980, −15.96770250721801793901234766778, −15.39405087868831194755776627848, −14.84928170684862651142673926107, −13.953583588071706061316558654, −13.22829607625190339313403950954, −12.22110202848224699916183243062, −11.22327320807450457253993614950, −10.43169103938483811879158371466, −9.826846590006631032679945785499, −9.148194368084005161583191833124, −7.286147892702348819551125531098, −6.86780239447430448810491246630, −5.51484089806313427676885437525, −4.67525623642993649432513444118, −3.75982788088024947521210898939, −2.664157924008372033598969973709, −2.540969990739302159772785939000, −0.26879593346990864159646044107, 1.1728280058750617141964477646, 2.52723406465184069600503481665, 3.33424921277518523847883708319, 4.41900891439958497908858119655, 5.59714157259142392555371152997, 6.28617064057869915172373833724, 7.13628204853710477894077480934, 8.20539333860974563094305690262, 8.70314476464593376344946855492, 9.81150984271734691407676009509, 11.452241554041650668778451559138, 12.198689787913748810828010162725, 12.849471547017254913479965593828, 13.43184988384127565023953711442, 14.14830959430065334319508320860, 15.19827958845689699392785982459, 16.02181701405369032753804508633, 16.85747153873774642052442600853, 17.38011284671063954601040317765, 18.7457375760795285383332373904, 19.533268819731583894773377821179, 20.10453171683104543161256134289, 21.17051002346002706313871566273, 21.77784921899397047531874130804, 22.921040490220254187892503102791

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