Properties

Label 1-920-920.147-r0-0-0
Degree $1$
Conductor $920$
Sign $0.0406 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.0406 + 0.999i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.0406 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.239252265 + 1.189819821i\)
\(L(\frac12)\) \(\approx\) \(1.239252265 + 1.189819821i\)
\(L(1)\) \(\approx\) \(1.260827052 + 0.3760410684i\)
\(L(1)\) \(\approx\) \(1.260827052 + 0.3760410684i\)

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

−21.748652931139329590526107013478, −20.74069217656867785226308660058, −19.97018840921275675557590744381, −19.333400866758648997875301159954, −18.97512575668361559381469553110, −17.6634941500510760818915911796, −16.95350652844703592022378069004, −16.021074991111959716941070703483, −15.29837923901961077770832883651, −14.33290890812079587582336393990, −13.70818539538834184019576822622, −13.109405527693897606351041248122, −12.120634551489905188937882845344, −11.159720857066445206525892262610, −10.097605250988251154007924798729, −9.26481241934172251035598850967, −8.827102220518205836578708092942, −7.56458270206384213348514280047, −6.91597213530211232981252868791, −6.183041672832162316530258254769, −4.506075677855423957350304093612, −3.984999199794983142477130425419, −2.882962570978622124071030753397, −2.07494091272834784074189167584, −0.64497563433690816458440676761, 1.55201144406058459155163667628, 2.49058462781167367993205494901, 3.358590946328343081213068741656, 4.248270098680575966250090799931, 5.296863162080463617121991686295, 6.55228325928686994871699607726, 7.15544024201883535161874668515, 8.44386534444148662997335155616, 8.83352305381714301988856641538, 9.93171430106197712279024543157, 10.2952694500580809673837673455, 11.80965861705733043837279178569, 12.588608020783663013000421353341, 13.14819295451232396503891947751, 14.284973843305142874326416752466, 14.88733567240051309859702904930, 15.531732837233788642966948709785, 16.366751141062192042023639349460, 17.35835817863304398360286429154, 18.258377329292394703279324992732, 19.16272829640591825090288353336, 19.740615356119942024907229279371, 20.27195813272533070914854253313, 21.31995585763044108810325795844, 22.08670213557647225727374676206

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