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Properties

Label 1-680-680.523-r0-0-0
Degree $1$
Conductor $680$
Sign $0.346 - 0.937i$
Analytic cond. $3.15790$
Root an. cond. $3.15790$
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  − 3-s + 7-s + 9-s − i·11-s − i·13-s + 19-s − 21-s − 23-s − 27-s + i·29-s − i·31-s + i·33-s − 37-s + i·39-s − i·41-s + ⋯
L(s)  = 1  − 3-s + 7-s + 9-s − i·11-s − i·13-s + 19-s − 21-s − 23-s − 27-s + i·29-s − i·31-s + i·33-s − 37-s + i·39-s − i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.346 - 0.937i$
Analytic conductor: \(3.15790\)
Root analytic conductor: \(3.15790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (0:\ ),\ 0.346 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8557475851 - 0.5958610731i\)
\(L(\frac12)\) \(\approx\) \(0.8557475851 - 0.5958610731i\)
\(L(1)\) \(\approx\) \(0.8630672070 - 0.1765596926i\)
\(L(1)\) \(\approx\) \(0.8630672070 - 0.1765596926i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
13 \( 1 \)
19 \( 1 + T \)
23 \( 1 \)
29 \( 1 + T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 \)
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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

−22.96014151696852588850978412161, −22.0942237331739192614473374049, −21.32971135664219133508019034402, −20.64223904547135113784826962660, −19.62331470289611221717976337989, −18.49830584085125412789377762427, −17.88285309617003531883504647420, −17.31810287256225906163809701917, −16.34729600496306585412734490470, −15.62799278789867903229937583135, −14.61205712390440944450727013466, −13.81465909332051450762672829591, −12.687400080197835993458826892439, −11.71779537944226917809523704453, −11.49721093570378506892979154248, −10.23809555700394482810531565967, −9.62993760547918011452212400720, −8.31975945588761205346904361196, −7.334101647371517776735455985164, −6.611107362415511184890233470569, −5.440143390274081317987755458359, −4.727975978066337814811865364839, −3.92645685062218468108535244874, −2.12459115958700312178588790839, −1.2977819313864929308072407307, 0.646802113809041213200469767749, 1.75649262792744364828046318844, 3.26650895151819965216094454140, 4.385499032722467615170573850057, 5.47170226533864130231643879785, 5.81975703395037801057127380027, 7.19888589876619274988111153646, 7.934905300993800549318133337849, 8.96432234876426986313024739773, 10.253455052599813014955778782534, 10.82469029860791477435143166705, 11.6725872161654926481817609652, 12.31526581766121362825059189275, 13.44235980073667610826733161836, 14.20027867933164282301550533911, 15.34273070906546484682028846513, 16.03465500654641625044889465277, 16.90714853668755338672016189065, 17.73193611299030525737456357912, 18.23601947146395996940963344490, 19.11492962846611725711568610530, 20.350633655282052280782678369077, 20.95810503715195109780600773228, 22.075876511875223474831007435385, 22.31292396285236000886424856476

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