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Properties

Label 1-709-709.25-r0-0-0
Degree $1$
Conductor $709$
Sign $0.681 - 0.731i$
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  + (0.842 − 0.537i)2-s + (−0.954 + 0.297i)3-s + (0.421 − 0.906i)4-s + (0.781 − 0.624i)5-s + (−0.645 + 0.764i)6-s + (−0.305 + 0.952i)7-s + (−0.132 − 0.991i)8-s + (0.823 − 0.567i)9-s + (0.322 − 0.946i)10-s + (−0.917 + 0.396i)11-s + (−0.132 + 0.991i)12-s + (0.977 + 0.211i)13-s + (0.254 + 0.967i)14-s + (−0.560 + 0.828i)15-s + (−0.645 − 0.764i)16-s + (0.115 + 0.993i)17-s + ⋯
L(s)  = 1  + (0.842 − 0.537i)2-s + (−0.954 + 0.297i)3-s + (0.421 − 0.906i)4-s + (0.781 − 0.624i)5-s + (−0.645 + 0.764i)6-s + (−0.305 + 0.952i)7-s + (−0.132 − 0.991i)8-s + (0.823 − 0.567i)9-s + (0.322 − 0.946i)10-s + (−0.917 + 0.396i)11-s + (−0.132 + 0.991i)12-s + (0.977 + 0.211i)13-s + (0.254 + 0.967i)14-s + (−0.560 + 0.828i)15-s + (−0.645 − 0.764i)16-s + (0.115 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.681 - 0.731i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.825630202 - 0.7948045123i\)
\(L(\frac12)\) \(\approx\) \(1.825630202 - 0.7948045123i\)
\(L(1)\) \(\approx\) \(1.405064156 - 0.4226876673i\)
\(L(1)\) \(\approx\) \(1.405064156 - 0.4226876673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.842 - 0.537i)T \)
3 \( 1 + (-0.954 + 0.297i)T \)
5 \( 1 + (0.781 - 0.624i)T \)
7 \( 1 + (-0.305 + 0.952i)T \)
11 \( 1 + (-0.917 + 0.396i)T \)
13 \( 1 + (0.977 + 0.211i)T \)
17 \( 1 + (0.115 + 0.993i)T \)
19 \( 1 + (0.924 + 0.380i)T \)
23 \( 1 + (0.355 - 0.934i)T \)
29 \( 1 + (0.603 - 0.797i)T \)
31 \( 1 + (0.997 + 0.0709i)T \)
37 \( 1 + (-0.305 + 0.952i)T \)
41 \( 1 + (0.969 + 0.245i)T \)
43 \( 1 + (0.758 + 0.651i)T \)
47 \( 1 + (-0.987 + 0.159i)T \)
53 \( 1 + (-0.437 - 0.899i)T \)
59 \( 1 + (-0.132 - 0.991i)T \)
61 \( 1 + (-0.973 - 0.228i)T \)
67 \( 1 + (-0.405 - 0.914i)T \)
71 \( 1 + (0.322 + 0.946i)T \)
73 \( 1 + (0.949 - 0.314i)T \)
79 \( 1 + (0.999 + 0.0354i)T \)
83 \( 1 + (0.910 + 0.413i)T \)
89 \( 1 + (-0.167 - 0.985i)T \)
97 \( 1 + (0.959 - 0.280i)T \)
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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

−22.92386901071342628973791933591, −22.17997370113417915114132611626, −21.228671971873640503735137484546, −20.76132732281476762455007270168, −19.43080958789018849703218959427, −18.137832469549047137092231214771, −17.867243110955058189067122585818, −16.911737951669044881077916476135, −16.037400311848518775305111912729, −15.6341242776560895553167224176, −14.10261991659610175672168731672, −13.57859349836205907079149811204, −13.136729480766463105419598468869, −11.992260352187109354942291157442, −10.95165387317006337970196177286, −10.607063534958790369920043733253, −9.32830489450245567656417569281, −7.71659520429177438414535558982, −7.18520445212158520804697150660, −6.33088062407567042890768380520, −5.579214440657416108388906044817, −4.860486242984747131445942363, −3.53690275219321577254111740868, −2.6645772659736698931668148226, −1.10512871439915253306196960237, 1.04972028107627202424296453929, 2.08796260459139319700824412871, 3.245191198146143268346612502103, 4.56834048040708390870704354216, 5.13086372571114477604779821578, 6.155649790827731506192729881500, 6.31996263457397082065041175824, 8.19215440917036344858236883273, 9.453806444593641516362716901319, 10.0538193448210584101720917136, 10.88034528440925859844728607461, 11.85081234160174683172061789751, 12.613568938370271371870921713471, 13.040789179494967300391982937174, 14.07999115681823274939476706768, 15.27025222749250816808911960590, 15.84835024498689504573321056742, 16.55863410265985808731548454857, 17.77436172457821804707517079475, 18.386052656574494571904914962267, 19.22873337301996201869257126731, 20.59880236223409177828408155712, 21.10904091406571787047472645288, 21.535134656283748961656513693077, 22.58441184091449533422626094083

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