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Properties

Label 1-709-709.181-r0-0-0
Degree $1$
Conductor $709$
Sign $0.718 - 0.695i$
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  + (−0.339 − 0.940i)2-s + (−0.987 − 0.159i)3-s + (−0.769 + 0.638i)4-s + (0.0797 − 0.996i)5-s + (0.185 + 0.982i)6-s + (0.658 + 0.752i)7-s + (0.861 + 0.507i)8-s + (0.949 + 0.314i)9-s + (−0.964 + 0.263i)10-s + (−0.833 − 0.552i)11-s + (0.861 − 0.507i)12-s + (0.658 − 0.752i)13-s + (0.484 − 0.874i)14-s + (−0.237 + 0.971i)15-s + (0.185 − 0.982i)16-s + (−0.833 + 0.552i)17-s + ⋯
L(s)  = 1  + (−0.339 − 0.940i)2-s + (−0.987 − 0.159i)3-s + (−0.769 + 0.638i)4-s + (0.0797 − 0.996i)5-s + (0.185 + 0.982i)6-s + (0.658 + 0.752i)7-s + (0.861 + 0.507i)8-s + (0.949 + 0.314i)9-s + (−0.964 + 0.263i)10-s + (−0.833 − 0.552i)11-s + (0.861 − 0.507i)12-s + (0.658 − 0.752i)13-s + (0.484 − 0.874i)14-s + (−0.237 + 0.971i)15-s + (0.185 − 0.982i)16-s + (−0.833 + 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7278645503 - 0.2946775958i\)
\(L(\frac12)\) \(\approx\) \(0.7278645503 - 0.2946775958i\)
\(L(1)\) \(\approx\) \(0.6225974046 - 0.2962447948i\)
\(L(1)\) \(\approx\) \(0.6225974046 - 0.2962447948i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.339 - 0.940i)T \)
3 \( 1 + (-0.987 - 0.159i)T \)
5 \( 1 + (0.0797 - 0.996i)T \)
7 \( 1 + (0.658 + 0.752i)T \)
11 \( 1 + (-0.833 - 0.552i)T \)
13 \( 1 + (0.658 - 0.752i)T \)
17 \( 1 + (-0.833 + 0.552i)T \)
19 \( 1 + (0.861 + 0.507i)T \)
23 \( 1 + (0.802 + 0.596i)T \)
29 \( 1 + (-0.0266 + 0.999i)T \)
31 \( 1 + (-0.237 + 0.971i)T \)
37 \( 1 + (0.658 + 0.752i)T \)
41 \( 1 + (-0.998 - 0.0532i)T \)
43 \( 1 + (0.734 - 0.678i)T \)
47 \( 1 + (0.802 + 0.596i)T \)
53 \( 1 + (-0.237 + 0.971i)T \)
59 \( 1 + (0.861 + 0.507i)T \)
61 \( 1 + (0.388 + 0.921i)T \)
67 \( 1 + (0.910 - 0.413i)T \)
71 \( 1 + (-0.964 - 0.263i)T \)
73 \( 1 + (0.288 + 0.957i)T \)
79 \( 1 + (-0.617 - 0.786i)T \)
83 \( 1 + (-0.132 - 0.991i)T \)
89 \( 1 + (-0.931 + 0.364i)T \)
97 \( 1 + (0.574 - 0.818i)T \)
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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

−22.85692293386520437917186496203, −22.292940499485788980184596250766, −21.25906138053966696430886658131, −20.33475352168510966113713178978, −18.959338760391577110992353041775, −18.31546006128583707758197817007, −17.774096170444007774567697348676, −17.11435397868482916226979849964, −16.1405318296949188740367534267, −15.48962114060521703271391611415, −14.69531451525032664439986856754, −13.71160828098056152538848356919, −13.09170154402594396422275790497, −11.3993348158201113506400164835, −11.04459008305930015552734748775, −10.11593641123022843339605339435, −9.39155972308501071928979718614, −7.97682348080098537451195554649, −7.10587595151706245150207632797, −6.70797063871180126337530661616, −5.60807289032408637737450358193, −4.7161759659082585075190386966, −3.95531937080455603394494593593, −2.11612037909047243349087558652, −0.6499614417384089564769892829, 0.98278654743636341719899387455, 1.69554194764948254681755145025, 3.067934904874161395889764330707, 4.373015570445609590189965407626, 5.30164888940046477468045522621, 5.694539538772719100779849323248, 7.43744630499376521707870516396, 8.3844345493538685882078905403, 8.94962874849023728683962841344, 10.176121256848545744451740414542, 10.940749099654232265678114898319, 11.605591181786844694519082258147, 12.483901052756550292790957803916, 13.00684915746307673306830139114, 13.8007100470219035816165700127, 15.44110306960401190850369459572, 16.12718264109561852889782224609, 17.05878592199104610148369949385, 17.81512326546147190464629889032, 18.30377230273221468185295832943, 19.124165917681865194070851843920, 20.29354868130127402973023860017, 20.87270478340920174117054264825, 21.68262954049096618491366647386, 22.17496781092664611122530787185

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