Properties

Degree 1
Conductor 709
Sign $0.340 - 0.940i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.0266 + 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,χ)  = 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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.340 - 0.940i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.340 - 0.940i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(709\)
\( \varepsilon \)  =  $0.340 - 0.940i$
motivic weight  =  \(0\)
character  :  $\chi_{709} (59, \cdot )$
Sato-Tate  :  $\mu(59)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 709,\ (0:\ ),\ 0.340 - 0.940i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8008066785 - 0.5615296276i$
$L(\frac12,\chi)$  $\approx$  $0.8008066785 - 0.5615296276i$
$L(\chi,1)$  $\approx$  0.9012875153 - 0.03917841624i
$L(1,\chi)$  $\approx$  0.9012875153 - 0.03917841624i

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

−22.606026674924443192684934589467, −21.54555749263491049090968414345, −21.34402668287782347526527101478, −20.60452383165550253262713048849, −19.57705958538357026686727587747, −18.860343821232137183482225277345, −17.90371027073530027890568136456, −17.085968345216756721848284489329, −16.51987409179556521186343261691, −15.194563649438516543712947000049, −14.207423168487930464592718618944, −13.86047025584352868185516353055, −12.88425720592287189066431088964, −11.47162364642875262459696158160, −10.89853948557685145372123423954, −10.452970685518611404650745191799, −9.5701042943213733759056705363, −8.80404122690657429257923998448, −7.65410044734513204892479562532, −6.21340710540587167035047142713, −5.38657443216177996152322090497, −4.17886729434226565396637115689, −3.62348276914544017348075143546, −2.60192096963479996246221389047, −1.38895109644055493493961523483, 0.497921814797503031479171254337, 1.86623278627541803906636625739, 2.91552211323048100534660204068, 5.03905063274927501489177498446, 5.10020770734959580472304066540, 6.18732411864802199842895092812, 7.06324062395158663296702537599, 7.940252555744687308918469022179, 8.839508511547404884517924691517, 9.32545944856494031210619376899, 10.67607124126746530800709923337, 12.008459838842331651225587692353, 12.93949041852880140260856408675, 13.104256268024140654606781588818, 14.18934900984160293829742929367, 15.16160448973132177570162697757, 15.82360758915879112066242387445, 16.944684655213763094306744804860, 17.603778555299724859112281552949, 18.1911794197653972161617101028, 18.71850280596039958130784231963, 20.04470261101082517907382389931, 20.7155627989148347974265824368, 22.09845609976854678865913712075, 22.45326402819413926030927226858

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