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Properties

Label 1-709-709.17-r1-0-0
Degree $1$
Conductor $709$
Sign $-0.962 + 0.270i$
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  + (0.651 + 0.758i)2-s + (−0.917 − 0.396i)3-s + (−0.150 + 0.988i)4-s + (0.746 + 0.665i)5-s + (−0.297 − 0.954i)6-s + (0.710 + 0.703i)7-s + (−0.847 + 0.530i)8-s + (0.684 + 0.728i)9-s + (−0.0177 + 0.999i)10-s + (0.271 − 0.962i)11-s + (0.530 − 0.847i)12-s + (0.263 − 0.964i)13-s + (−0.0709 + 0.997i)14-s + (−0.421 − 0.906i)15-s + (−0.954 − 0.297i)16-s + (−0.245 + 0.969i)17-s + ⋯
L(s)  = 1  + (0.651 + 0.758i)2-s + (−0.917 − 0.396i)3-s + (−0.150 + 0.988i)4-s + (0.746 + 0.665i)5-s + (−0.297 − 0.954i)6-s + (0.710 + 0.703i)7-s + (−0.847 + 0.530i)8-s + (0.684 + 0.728i)9-s + (−0.0177 + 0.999i)10-s + (0.271 − 0.962i)11-s + (0.530 − 0.847i)12-s + (0.263 − 0.964i)13-s + (−0.0709 + 0.997i)14-s + (−0.421 − 0.906i)15-s + (−0.954 − 0.297i)16-s + (−0.245 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.962 + 0.270i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.962 + 0.270i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3294770875 + 2.389752107i\)
\(L(\frac12)\) \(\approx\) \(0.3294770875 + 2.389752107i\)
\(L(1)\) \(\approx\) \(1.053803819 + 0.8556221412i\)
\(L(1)\) \(\approx\) \(1.053803819 + 0.8556221412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.651 + 0.758i)T \)
3 \( 1 + (-0.917 - 0.396i)T \)
5 \( 1 + (0.746 + 0.665i)T \)
7 \( 1 + (0.710 + 0.703i)T \)
11 \( 1 + (0.271 - 0.962i)T \)
13 \( 1 + (0.263 - 0.964i)T \)
17 \( 1 + (-0.245 + 0.969i)T \)
19 \( 1 + (-0.468 + 0.883i)T \)
23 \( 1 + (0.998 + 0.0620i)T \)
29 \( 1 + (-0.405 + 0.914i)T \)
31 \( 1 + (-0.0886 + 0.996i)T \)
37 \( 1 + (0.703 - 0.710i)T \)
41 \( 1 + (0.952 + 0.305i)T \)
43 \( 1 + (-0.631 - 0.775i)T \)
47 \( 1 + (0.833 + 0.552i)T \)
53 \( 1 + (0.818 + 0.574i)T \)
59 \( 1 + (-0.530 - 0.847i)T \)
61 \( 1 + (0.476 + 0.879i)T \)
67 \( 1 + (-0.792 - 0.610i)T \)
71 \( 1 + (-0.0177 - 0.999i)T \)
73 \( 1 + (-0.921 + 0.388i)T \)
79 \( 1 + (0.999 + 0.0443i)T \)
83 \( 1 + (-0.507 + 0.861i)T \)
89 \( 1 + (0.567 + 0.823i)T \)
97 \( 1 + (0.347 + 0.937i)T \)
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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

−21.92927956767332581500210356122, −21.1752034099373078075499350670, −20.69833519051176044185029176320, −20.02104821358963546517746479037, −18.73470702278283976258249324213, −17.85000487041417320991875527714, −17.21333766373060984117003361812, −16.447738879016265425253966410687, −15.30253971280308888557328592811, −14.54039420602926655474841766075, −13.45322266938135991318726890007, −13.00586129148971705671523278863, −11.77305155182786322963571517148, −11.40942211508801681249442925254, −10.42985836949843350642608573599, −9.58826915462431825399670672197, −9.03009999047698179951118093034, −7.16643535930456460555180029532, −6.36125323802473118267047434124, −5.29555193700030563281931374150, −4.49953781024542228888361926927, −4.223132041769681634567884651238, −2.39150475933351821099732956396, −1.401962181123331389239190910693, −0.52168060352159246323700696205, 1.33464991370122634191170138963, 2.5567830255757857334983392447, 3.69925825331368529746213033371, 5.05504156718152179713974758492, 5.79911105355433857203988144454, 6.15114815534730324479344500711, 7.2154707219205327237421795214, 8.16394835078905007920574261009, 9.02981710427719488801059205515, 10.691724067212645879682080325296, 11.01291634317053959714967581844, 12.22374695630628162912442293995, 12.8650160019545613430051729711, 13.70784078213519281648071557459, 14.6264395557876062814899071996, 15.22791516177617974574706907996, 16.33551318796254139513643628497, 17.07352027359478326291302671197, 17.80661754286861391286967631471, 18.32205513583782471479073504311, 19.19916039182823911461621017723, 20.83574466754956795646371500342, 21.67564633614024631384323587982, 21.898556465509526836867112418483, 22.84967222043252066140438027450

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