Properties

Label 1-1305-1305.212-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.553 - 0.832i$
Analytic cond. $6.06039$
Root an. cond. $6.06039$
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  + (−0.930 + 0.365i)2-s + (0.733 − 0.680i)4-s + (0.680 − 0.733i)7-s + (−0.433 + 0.900i)8-s + (0.0747 − 0.997i)11-s + (0.563 − 0.826i)13-s + (−0.365 + 0.930i)14-s + (0.0747 − 0.997i)16-s + i·17-s + (−0.222 − 0.974i)19-s + (0.294 + 0.955i)22-s + (0.930 + 0.365i)23-s + (−0.222 + 0.974i)26-s − i·28-s + (0.988 + 0.149i)31-s + (0.294 + 0.955i)32-s + ⋯
L(s)  = 1  + (−0.930 + 0.365i)2-s + (0.733 − 0.680i)4-s + (0.680 − 0.733i)7-s + (−0.433 + 0.900i)8-s + (0.0747 − 0.997i)11-s + (0.563 − 0.826i)13-s + (−0.365 + 0.930i)14-s + (0.0747 − 0.997i)16-s + i·17-s + (−0.222 − 0.974i)19-s + (0.294 + 0.955i)22-s + (0.930 + 0.365i)23-s + (−0.222 + 0.974i)26-s − i·28-s + (0.988 + 0.149i)31-s + (0.294 + 0.955i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9786069521 - 0.5245546381i\)
\(L(\frac12)\) \(\approx\) \(0.9786069521 - 0.5245546381i\)
\(L(1)\) \(\approx\) \(0.8193243268 - 0.1014200099i\)
\(L(1)\) \(\approx\) \(0.8193243268 - 0.1014200099i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.930 + 0.365i)T \)
7 \( 1 + (0.680 - 0.733i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.563 - 0.826i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.222 - 0.974i)T \)
23 \( 1 + (0.930 + 0.365i)T \)
31 \( 1 + (0.988 + 0.149i)T \)
37 \( 1 + (-0.433 + 0.900i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.149 - 0.988i)T \)
47 \( 1 + (0.997 + 0.0747i)T \)
53 \( 1 + (0.781 + 0.623i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (-0.997 + 0.0747i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.781 + 0.623i)T \)
79 \( 1 + (0.826 - 0.563i)T \)
83 \( 1 + (-0.294 + 0.955i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (0.294 - 0.955i)T \)
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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

−20.99056337376784127438347469912, −20.41928848179028853124571346086, −19.44238580451762190353861005468, −18.63857218302140340918493209777, −18.22646673364636683494032165601, −17.441978812951711858801934293321, −16.6507266093205920058840456512, −15.89121677807415870359177219909, −15.09109177294552918245099084576, −14.34737344475996579887761273771, −13.20444969840734179001007863820, −12.222783250989682007357945392967, −11.76866793011697742736732996107, −10.998991509490904138476069578648, −10.08050086297769295986127355104, −9.279002949014938855045382167513, −8.66863782050933931784308340744, −7.81591468606196042767561473390, −7.001213293619148487052780709104, −6.162884337288639467412916887384, −4.95459257680903678064679612785, −4.05292371066235265682029851030, −2.81962986632765647209550778156, −2.02198676461078771562751559123, −1.19517985799332626345034346315, 0.69013028466607759459867135067, 1.43964103469817203178024419496, 2.724140288754572171597993854337, 3.74468339621831174935285264144, 5.00293942854867469029975537499, 5.806247470741291026160683268016, 6.716380947884038704849215684777, 7.48661620544641625956654294525, 8.51109493897491218134891024116, 8.65464657075766545954770771733, 10.0086294310373638643566107697, 10.77570106242384985455687332725, 11.08872169700638095043995323168, 12.13446163090654543528409507867, 13.44010367015149325382985189316, 13.89317435069211885864927620857, 15.11400790318860868074377007138, 15.41006106781254272817373576936, 16.47963049735583979095817323637, 17.29286152168851633355733818532, 17.504191417009095437190589723787, 18.60596822736950250779208803360, 19.22031675242052809843875987213, 19.961828289410973504983053313, 20.7248836793219896972394732996

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