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Properties

Label 1-920-920.883-r0-0-0
Degree $1$
Conductor $920$
Sign $0.911 + 0.410i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (−0.989 − 0.142i)3-s + (0.755 − 0.654i)7-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (−0.909 − 0.415i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (0.281 − 0.959i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (0.755 − 0.654i)7-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (−0.909 − 0.415i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (0.281 − 0.959i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.911 + 0.410i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.911 + 0.410i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263167451 + 0.2711847093i\)
\(L(\frac12)\) \(\approx\) \(1.263167451 + 0.2711847093i\)
\(L(1)\) \(\approx\) \(0.9753102731 + 0.03690780002i\)
\(L(1)\) \(\approx\) \(0.9753102731 + 0.03690780002i\)

Euler product

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

−21.88460451107857519629655205535, −21.21518090413376010463345131444, −20.47164656205590381270122531850, −19.225831344776225085280032733427, −18.59493488530621544735233339020, −17.776580846652015552953344391978, −17.11911465637604845545803703928, −16.43536788411155463020369361265, −15.34670363504894546594621576545, −14.973052022907002666940170975683, −13.718120785324982349658082853899, −12.91286399542340940436832748213, −11.79259340201024184538452027738, −11.548007186631031958468516044640, −10.60032483682791589084880156800, −9.74774126865087183769601098862, −8.65092254648272101974665783976, −7.95352808488949503777915231365, −6.6533017977545399330653785837, −5.976790408632317027306793876002, −5.22099715339339539295200326172, −4.290052277094615156423818070043, −3.26454022521490490696610515739, −1.807678111194969976156996953590, −0.79751330591153644173867807133, 1.19245968371022426629787197625, 1.693086970475995368376884821574, 3.552799670285415152226937161834, 4.39462339001001560129646432198, 5.16670338849654034887090776966, 6.239085177823925947360795687556, 6.95578106648654711781677872615, 7.77362701532311493038722451442, 8.84672119981271901811522857348, 9.9838696570687469834455431829, 10.63959733549690957896037317056, 11.522724379207853094005190640523, 12.08370219744176156490207908932, 12.954910201895968278128686720536, 14.10454830503988552824671098719, 14.52768478441758627724171082905, 15.81798533539138782194368456319, 16.59046554427900655382797173526, 17.09824959900638621661944404854, 17.96727259445562689392583833995, 18.557283266174950885842844760163, 19.50829505519280102048577556472, 20.512441328559925071673987528007, 21.208141995443203495387358757298, 21.92216546583652386558963120521

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