Properties

Degree $1$
Conductor $709$
Sign $-0.962 + 0.270i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.962 + 0.270i$
Motivic weight: \(0\)
Character: $\chi_{709} (17, \cdot )$
Sato-Tate group: $\mu(708)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.962 + 0.270i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

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