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Properties

Label 1-709-709.12-r0-0-0
Degree $1$
Conductor $709$
Sign $0.637 + 0.770i$
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  + (0.0266 + 0.999i)2-s + (−0.132 + 0.991i)3-s + (−0.998 + 0.0532i)4-s + (0.658 + 0.752i)5-s + (−0.994 − 0.106i)6-s + (0.185 − 0.982i)7-s + (−0.0797 − 0.996i)8-s + (−0.964 − 0.263i)9-s + (−0.734 + 0.678i)10-s + (−0.530 − 0.847i)11-s + (0.0797 − 0.996i)12-s + (−0.185 − 0.982i)13-s + (0.987 + 0.159i)14-s + (−0.833 + 0.552i)15-s + (0.994 − 0.106i)16-s + (0.530 − 0.847i)17-s + ⋯
L(s)  = 1  + (0.0266 + 0.999i)2-s + (−0.132 + 0.991i)3-s + (−0.998 + 0.0532i)4-s + (0.658 + 0.752i)5-s + (−0.994 − 0.106i)6-s + (0.185 − 0.982i)7-s + (−0.0797 − 0.996i)8-s + (−0.964 − 0.263i)9-s + (−0.734 + 0.678i)10-s + (−0.530 − 0.847i)11-s + (0.0797 − 0.996i)12-s + (−0.185 − 0.982i)13-s + (0.987 + 0.159i)14-s + (−0.833 + 0.552i)15-s + (0.994 − 0.106i)16-s + (0.530 − 0.847i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.035239323 + 0.4867407384i\)
\(L(\frac12)\) \(\approx\) \(1.035239323 + 0.4867407384i\)
\(L(1)\) \(\approx\) \(0.8128326207 + 0.5302863493i\)
\(L(1)\) \(\approx\) \(0.8128326207 + 0.5302863493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.0266 + 0.999i)T \)
3 \( 1 + (-0.132 + 0.991i)T \)
5 \( 1 + (0.658 + 0.752i)T \)
7 \( 1 + (0.185 - 0.982i)T \)
11 \( 1 + (-0.530 - 0.847i)T \)
13 \( 1 + (-0.185 - 0.982i)T \)
17 \( 1 + (0.530 - 0.847i)T \)
19 \( 1 + (0.0797 + 0.996i)T \)
23 \( 1 + (-0.861 - 0.507i)T \)
29 \( 1 + (-0.237 - 0.971i)T \)
31 \( 1 + (0.833 - 0.552i)T \)
37 \( 1 + (-0.185 + 0.982i)T \)
41 \( 1 + (0.887 - 0.461i)T \)
43 \( 1 + (0.910 + 0.413i)T \)
47 \( 1 + (0.861 + 0.507i)T \)
53 \( 1 + (0.833 - 0.552i)T \)
59 \( 1 + (0.0797 + 0.996i)T \)
61 \( 1 + (0.437 - 0.899i)T \)
67 \( 1 + (-0.769 - 0.638i)T \)
71 \( 1 + (-0.734 - 0.678i)T \)
73 \( 1 + (-0.484 - 0.874i)T \)
79 \( 1 + (-0.288 - 0.957i)T \)
83 \( 1 + (0.931 - 0.364i)T \)
89 \( 1 + (-0.977 - 0.211i)T \)
97 \( 1 + (0.697 - 0.716i)T \)
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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

−22.242168391427389347086128516536, −21.5923821882672870496033081498, −20.93073207071751422647099868163, −19.9470504068676608249935583188, −19.32147655527012460978154937952, −18.45042352461658834229282113634, −17.73447629008003888845941156787, −17.31790097430551872657827754711, −16.081786310156144239186865232504, −14.71802732491015862217131378677, −13.95586326827835451412013918929, −13.111500782813850666530999993164, −12.37550284348801823393083323560, −12.04451662231507679167686136310, −10.97191865279256789915305571213, −9.85074166176177652861318467568, −8.994380787134774967000478607634, −8.407150201066118557854224644132, −7.24054766083424816917518569405, −5.86360585422034021014078198846, −5.29505409865487578956954535251, −4.27331932059217291167836720399, −2.61236511451969947451282009359, −2.05345131820422793000186674781, −1.21931082481289291568122469967, 0.61526230435442301492263634606, 2.8196076677930603530890951404, 3.68062274996351175579278149550, 4.66972961638166127129367824566, 5.7379202025927322409971061072, 6.12840719621550234382038117793, 7.54199336978597508237271986471, 8.088448967798203137743913050149, 9.36389483126623978250230476427, 10.2695072263478097563193573845, 10.46135535276787333785377898767, 11.80090973458383091154237201610, 13.26262577484010077029978416093, 13.9774183476458241024816801577, 14.47509419925352456524959712134, 15.394847651174897317560584512354, 16.19974500685434682469041857017, 16.8890653986888362989289897035, 17.5986844643322533722845601116, 18.348869937137040100574052237337, 19.31075538493954873318140507784, 20.74824616578880701147547111952, 21.03949049536949121132001430192, 22.392605337391885025853683616452, 22.52724445673283749683758743549

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