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Properties

Label 1-709-709.697-r0-0-0
Degree $1$
Conductor $709$
Sign $0.340 + 0.940i$
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

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

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.340 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8008066785 + 0.5615296276i\)
\(L(\frac12)\) \(\approx\) \(0.8008066785 + 0.5615296276i\)
\(L(1)\) \(\approx\) \(0.9012875153 + 0.03917841624i\)
\(L(1)\) \(\approx\) \(0.9012875153 + 0.03917841624i\)

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

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

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