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Properties

Label 1-695-695.457-r0-0-0
Degree $1$
Conductor $695$
Sign $0.999 + 0.00237i$
Analytic cond. $3.22756$
Root an. cond. $3.22756$
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  + (0.789 − 0.613i)2-s + (0.356 + 0.934i)3-s + (0.247 − 0.968i)4-s + (0.854 + 0.519i)6-s + (−0.999 + 0.0227i)7-s + (−0.398 − 0.917i)8-s + (−0.746 + 0.665i)9-s + (0.995 + 0.0909i)11-s + (0.993 − 0.113i)12-s + (0.987 + 0.158i)13-s + (−0.775 + 0.631i)14-s + (−0.877 − 0.480i)16-s + (−0.0455 + 0.998i)17-s + (−0.181 + 0.983i)18-s + (0.538 − 0.842i)19-s + ⋯
L(s)  = 1  + (0.789 − 0.613i)2-s + (0.356 + 0.934i)3-s + (0.247 − 0.968i)4-s + (0.854 + 0.519i)6-s + (−0.999 + 0.0227i)7-s + (−0.398 − 0.917i)8-s + (−0.746 + 0.665i)9-s + (0.995 + 0.0909i)11-s + (0.993 − 0.113i)12-s + (0.987 + 0.158i)13-s + (−0.775 + 0.631i)14-s + (−0.877 − 0.480i)16-s + (−0.0455 + 0.998i)17-s + (−0.181 + 0.983i)18-s + (0.538 − 0.842i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 695 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 695 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(695\)    =    \(5 \cdot 139\)
Sign: $0.999 + 0.00237i$
Analytic conductor: \(3.22756\)
Root analytic conductor: \(3.22756\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{695} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 695,\ (0:\ ),\ 0.999 + 0.00237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.422837578 + 0.002875717152i\)
\(L(\frac12)\) \(\approx\) \(2.422837578 + 0.002875717152i\)
\(L(1)\) \(\approx\) \(1.744147024 - 0.1083828479i\)
\(L(1)\) \(\approx\) \(1.744147024 - 0.1083828479i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.789 - 0.613i)T \)
3 \( 1 + (0.356 + 0.934i)T \)
7 \( 1 + (-0.999 + 0.0227i)T \)
11 \( 1 + (0.995 + 0.0909i)T \)
13 \( 1 + (0.987 + 0.158i)T \)
17 \( 1 + (-0.0455 + 0.998i)T \)
19 \( 1 + (0.538 - 0.842i)T \)
23 \( 1 + (0.519 + 0.854i)T \)
29 \( 1 + (0.715 + 0.699i)T \)
31 \( 1 + (0.746 + 0.665i)T \)
37 \( 1 + (0.557 - 0.829i)T \)
41 \( 1 + (-0.648 - 0.761i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.956 + 0.291i)T \)
53 \( 1 + (0.926 + 0.377i)T \)
59 \( 1 + (0.962 + 0.269i)T \)
61 \( 1 + (0.648 - 0.761i)T \)
67 \( 1 + (-0.907 - 0.419i)T \)
71 \( 1 + (0.291 - 0.956i)T \)
73 \( 1 + (-0.595 + 0.803i)T \)
79 \( 1 + (0.334 + 0.942i)T \)
83 \( 1 + (-0.907 + 0.419i)T \)
89 \( 1 + (-0.898 - 0.439i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−22.80892865761148710805988314943, −22.30099436522163485145221264001, −20.92635830122788101047642484240, −20.36876834973608827500788714230, −19.414715144075463196087983194573, −18.56817684063701451763952622744, −17.74884567962787932848189915147, −16.6998745009851960084610086756, −16.1524481307101232496288362200, −15.08113318924396963580840305052, −14.263580721569795021666840374494, −13.48948718263833730121131441775, −13.02338460648058095600771621907, −11.95530278419894421849799803456, −11.5251859721039080185052468827, −9.84914585008630574876565687094, −8.81042288645907515181303975147, −8.084503752790051337307812485312, −6.98754861917671415188392401348, −6.42300463984294701142822023413, −5.764277799151069497369777187928, −4.28574365136175445689616914745, −3.31307279174045188983105515085, −2.618923527493902239752964170512, −1.04515305289742338567373661097, 1.25007556690723350968669261377, 2.65055788206571684126586873447, 3.558824399913766169330540970495, 4.02844994190558556432198733384, 5.19861182293823694342191698796, 6.121003872948836562468752205450, 6.96250347808683762474011855882, 8.71628550896177242747576004612, 9.29311398584616915154601897512, 10.16679358656312066933744200391, 10.939510428965032855885721705369, 11.75163381016535888952536930702, 12.78378412911782300464248598470, 13.63574449359358908410666231682, 14.26775019312621281482599068889, 15.30333257202949447819278631805, 15.78876158820974905677454166041, 16.63366117999323331522828423350, 17.77612418805607995527745196396, 19.188917496259386906279073665465, 19.56861467390943625338759347826, 20.23809516437654012223163338632, 21.27221905258285714817249555223, 21.74943859362175892485954356668, 22.53582762710917882765583370827

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