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Properties

Label 1-460-460.319-r0-0-0
Degree $1$
Conductor $460$
Sign $0.764 - 0.644i$
Analytic cond. $2.13623$
Root an. cond. $2.13623$
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  + (−0.654 − 0.755i)3-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.415 − 0.909i)13-s + (0.841 + 0.540i)17-s + (0.841 − 0.540i)19-s + (0.959 − 0.281i)21-s + (0.841 − 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + (−0.142 + 0.989i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)3-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.415 − 0.909i)13-s + (0.841 + 0.540i)17-s + (0.841 − 0.540i)19-s + (0.959 − 0.281i)21-s + (0.841 − 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + (−0.142 + 0.989i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8507321155 - 0.3106825745i\)
\(L(\frac12)\) \(\approx\) \(0.8507321155 - 0.3106825745i\)
\(L(1)\) \(\approx\) \(0.8029322319 - 0.1527503579i\)
\(L(1)\) \(\approx\) \(0.8029322319 - 0.1527503579i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.654 - 0.755i)T \)
7 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (-0.142 + 0.989i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + T \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−23.59406391062679600159374951932, −23.23133514685343593800167498870, −22.43292240259793759160638392741, −21.35848794225962366726277867464, −20.80904658832820829223973597607, −19.91143796849998601291794714447, −18.81154676432657394697777358362, −17.86367726770711711778317241046, −16.98186315791811176036313661840, −16.24116851035515657722005027776, −15.67730228515520846538792497406, −14.417387372337847720272872134539, −13.700510674944758162262889376305, −12.41057635956568476529072873051, −11.73286488946670181977411645150, −10.532203151344781957034061165548, −10.04950160253611908851882192862, −9.18558311619845261997743235180, −7.70880944504756614110404375406, −6.88153884721536477735357160603, −5.72297323094137450916783114570, −4.78878577730658585663992611678, −3.9037812004319280685423652828, −2.780550659568784059582876698013, −0.924238641730427289215878588026, 0.76556568662954663462274532087, 2.325397758852065726653295951235, 3.155440238609381252562186247, 5.070197606308270019688915727355, 5.58264659526505061552596969479, 6.56094364326268648155949587799, 7.71074031838605624860987657530, 8.371657095429717989003412710052, 9.778270041377358820051039644827, 10.62730969794374413762063192341, 11.69888655067593553985995733664, 12.48376430132162603050503164932, 13.08240982611110118758533245316, 14.10263721889529225640297648183, 15.395633193018379440150652195559, 15.99432685123841116773510735245, 17.10174489439690329205172129078, 17.89460307876066080836027780467, 18.67577139824954613893871359891, 19.26531403910055294248971470019, 20.34670547334851171794332573779, 21.5042208106722890624500501308, 22.23615573380949846219247493890, 22.964969472885138510689133075, 23.88346044611933931699348120064

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