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Properties

Label 1-709-709.106-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.122 + 0.992i$
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  + (0.746 + 0.665i)2-s + (0.879 − 0.476i)3-s + (0.115 + 0.993i)4-s + (−0.271 + 0.962i)5-s + (0.973 + 0.228i)6-s + (0.989 + 0.141i)7-s + (−0.574 + 0.818i)8-s + (0.545 − 0.838i)9-s + (−0.842 + 0.537i)10-s + (−0.813 − 0.582i)11-s + (0.574 + 0.818i)12-s + (0.617 + 0.786i)13-s + (0.645 + 0.764i)14-s + (0.220 + 0.975i)15-s + (−0.973 + 0.228i)16-s + (0.0974 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.746 + 0.665i)2-s + (0.879 − 0.476i)3-s + (0.115 + 0.993i)4-s + (−0.271 + 0.962i)5-s + (0.973 + 0.228i)6-s + (0.989 + 0.141i)7-s + (−0.574 + 0.818i)8-s + (0.545 − 0.838i)9-s + (−0.842 + 0.537i)10-s + (−0.813 − 0.582i)11-s + (0.574 + 0.818i)12-s + (0.617 + 0.786i)13-s + (0.645 + 0.764i)14-s + (0.220 + 0.975i)15-s + (−0.973 + 0.228i)16-s + (0.0974 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.122 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.947464768 + 2.201685709i\)
\(L(\frac12)\) \(\approx\) \(1.947464768 + 2.201685709i\)
\(L(1)\) \(\approx\) \(1.781890526 + 1.019155149i\)
\(L(1)\) \(\approx\) \(1.781890526 + 1.019155149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.746 + 0.665i)T \)
3 \( 1 + (0.879 - 0.476i)T \)
5 \( 1 + (-0.271 + 0.962i)T \)
7 \( 1 + (0.989 + 0.141i)T \)
11 \( 1 + (-0.813 - 0.582i)T \)
13 \( 1 + (0.617 + 0.786i)T \)
17 \( 1 + (0.0974 + 0.995i)T \)
19 \( 1 + (-0.996 - 0.0886i)T \)
23 \( 1 + (0.405 + 0.914i)T \)
29 \( 1 + (0.710 - 0.703i)T \)
31 \( 1 + (0.954 + 0.297i)T \)
37 \( 1 + (-0.989 - 0.141i)T \)
41 \( 1 + (0.870 - 0.492i)T \)
43 \( 1 + (-0.992 + 0.123i)T \)
47 \( 1 + (0.994 + 0.106i)T \)
53 \( 1 + (-0.734 + 0.678i)T \)
59 \( 1 + (0.574 - 0.818i)T \)
61 \( 1 + (0.981 + 0.194i)T \)
67 \( 1 + (-0.560 - 0.828i)T \)
71 \( 1 + (-0.842 - 0.537i)T \)
73 \( 1 + (-0.977 - 0.211i)T \)
79 \( 1 + (-0.150 + 0.988i)T \)
83 \( 1 + (0.237 - 0.971i)T \)
89 \( 1 + (-0.895 - 0.445i)T \)
97 \( 1 + (-0.355 + 0.934i)T \)
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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

−22.26587644821107836262552937257, −21.06108094219621233978790831848, −20.81063535257231557702977242226, −20.39920959804039235447270384085, −19.493271725366353400145956410327, −18.57429769829461895269851708881, −17.641024390629481016824474712772, −16.29618212377049072113867886532, −15.56975250155876820709252962435, −14.92441628835030654491135147678, −14.06812185539846390834956935199, −13.22562147946920061334015303897, −12.64546251411056750008743880962, −11.61081105051436272718134722243, −10.5999496648627091779209812945, −10.03711604709385042441033813842, −8.77842259889865370129527602810, −8.27099610870151922180605144726, −7.15120279156093321669874117288, −5.50256368074849210584321436399, −4.73682558562858093875797790427, −4.29858935988457206548038466121, −3.04973113553678057707308511566, −2.15019164010372266012138968119, −1.03669065438004542668278437382, 1.8159426367045185366645937802, 2.72380573354234652948399347128, 3.65741330991045447344863833009, 4.4697141905836915242787040716, 5.86420504650826753479517340172, 6.61586014323139566949553157293, 7.547124309866133013866005632639, 8.22762070502928490028457760057, 8.80808247368798181655871086049, 10.45538725673370598871739423087, 11.33426450895367848025223385724, 12.17007970456272852257140321632, 13.26101983313050058437664943810, 13.91473395787780284894106096540, 14.50171641546622169931403657703, 15.3382600697557182585173324609, 15.75329427512445850204632238306, 17.24791889354547034896230887468, 17.878026194854835745555772494627, 18.86634354988680746059214971388, 19.352850999192063333265708969011, 20.81269591245271021510512306163, 21.25518463744071152384211539787, 21.826540819358806985621999781400, 23.34437777957463344370907807268

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