Properties

Degree 1
Conductor $ 17 \cdot 43 $
Sign $0.119 - 0.992i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.707 + 0.707i)2-s + (−0.991 − 0.130i)3-s + i·4-s + (−0.130 + 0.991i)5-s + (−0.608 − 0.793i)6-s + (−0.130 − 0.991i)7-s + (−0.707 + 0.707i)8-s + (0.965 + 0.258i)9-s + (−0.793 + 0.608i)10-s + (0.382 − 0.923i)11-s + (0.130 − 0.991i)12-s + (−0.866 + 0.5i)13-s + (0.608 − 0.793i)14-s + (0.258 − 0.965i)15-s − 16-s + ⋯
L(s,χ)  = 1  + (0.707 + 0.707i)2-s + (−0.991 − 0.130i)3-s + i·4-s + (−0.130 + 0.991i)5-s + (−0.608 − 0.793i)6-s + (−0.130 − 0.991i)7-s + (−0.707 + 0.707i)8-s + (0.965 + 0.258i)9-s + (−0.793 + 0.608i)10-s + (0.382 − 0.923i)11-s + (0.130 − 0.991i)12-s + (−0.866 + 0.5i)13-s + (0.608 − 0.793i)14-s + (0.258 − 0.965i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.119 - 0.992i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.119 - 0.992i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $0.119 - 0.992i$
motivic weight  =  \(0\)
character  :  $\chi_{731} (7, \cdot )$
Sato-Tate  :  $\mu(48)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 731,\ (0:\ ),\ 0.119 - 0.992i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1708849090 - 0.1515376215i$
$L(\frac12,\chi)$  $\approx$  $0.1708849090 - 0.1515376215i$
$L(\chi,1)$  $\approx$  0.7254034948 + 0.3178299388i
$L(1,\chi)$  $\approx$  0.7254034948 + 0.3178299388i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−22.64547717135050930803208116547, −21.78639175470755771044347451527, −21.409924598661354696069303011510, −20.363703969372684925625757189833, −19.64680727302053671717708324052, −18.810056026865714583453623907759, −17.71649086737908002638039861734, −17.19313474651113983411517434425, −15.92119823427789328715885496635, −15.42514504436823081329975749481, −14.60074123003455464653431474404, −13.16554954887026249427933909988, −12.58232426897396821924589576267, −12.042642963279223318848965929, −11.46626179283648535677240721432, −10.17158576550037456866399334615, −9.65350386770995477085802360597, −8.66423599231631128439743015393, −7.17601939234467837308327146772, −6.0950391783507609812248714963, −5.314634956685098916878727091, −4.72007186751263522390984304199, −3.85594633923784970053941927674, −2.33353501190408101277992970701, −1.4304015393959030226809293712, 0.09381530014907817882393885911, 2.093908026553054192253461189438, 3.51817560883673747247743940305, 4.17359009064207585849326713422, 5.22459647064538369226454741783, 6.31676963259140396456656011735, 6.79047834366396040149770892888, 7.445000858380615089869639676289, 8.57831455297339280050704974104, 10.12140864466375420062991361752, 10.78431995563289835301955211879, 11.66110921870487009505969400660, 12.38380326331899896827811408514, 13.42369846914787239737088976630, 14.17537500149494383156176140934, 14.825351957628792744916527608688, 15.956116811606269200205609579893, 16.66080976115272339963207283080, 17.15550207189604620032793363712, 18.07199985878645977758033780706, 18.93196889980490654432461712002, 19.853620465971172070873137544699, 21.20812798813726252737662275298, 21.872816345896852477946090145077, 22.432724191609181221713845020142

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