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Properties

Label 1-920-920.107-r1-0-0
Degree $1$
Conductor $920$
Sign $0.0313 + 0.999i$
Analytic cond. $98.8677$
Root an. cond. $98.8677$
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  + (0.755 + 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (0.959 − 0.281i)11-s + (0.909 + 0.415i)13-s + (−0.540 − 0.841i)17-s + (0.841 + 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.540 + 0.841i)27-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (0.989 − 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (0.959 − 0.281i)11-s + (0.909 + 0.415i)13-s + (−0.540 − 0.841i)17-s + (0.841 + 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.540 + 0.841i)27-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (0.989 − 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.0313 + 0.999i$
Analytic conductor: \(98.8677\)
Root analytic conductor: \(98.8677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (1:\ ),\ 0.0313 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.093995356 + 2.029397297i\)
\(L(\frac12)\) \(\approx\) \(2.093995356 + 2.029397297i\)
\(L(1)\) \(\approx\) \(1.357717667 + 0.5028637672i\)
\(L(1)\) \(\approx\) \(1.357717667 + 0.5028637672i\)

Euler product

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

−21.45057851528419322288938845929, −20.3521668021012937862952648084, −19.89947737828667643374649788290, −19.3180773190309393734714956301, −18.39909798283560852173840080633, −17.65340627284148264323130088994, −16.81529974124969426153469392103, −15.75220745170378809404997610150, −15.13645285846107461955475737774, −14.09010716952321929106893202528, −13.48344746574719293896433416393, −12.807469897385590399870052766630, −12.00639152673566492076527259272, −10.96850965142987374202634593866, −9.88321899844453103303164236563, −9.1744466969908714382763370219, −8.38471894456292145323235299818, −7.4277189759584121905189682230, −6.555081311467504790502670830122, −6.063112486181543370886550524302, −4.38614691319051123095991494009, −3.55605258291755973994161162912, −2.78789005474389216791802265384, −1.50496391499767367395347888424, −0.658104630126861162464807099385, 1.047973836668059111356767274270, 2.379350848841834541652249881331, 3.29507224028135076756340807014, 3.96044637948241411632597462302, 5.02642719419800271336439686303, 6.16728835739279493956019091174, 6.91568379735068057944924254936, 8.17558072675852407384439835899, 8.92747920434152353864744314957, 9.5293289354001327012901531407, 10.281285863597164830506678749481, 11.420648097359849887698833390362, 12.093483296131584092015233985842, 13.45103495840199932018873505092, 13.745439329403526780815615608963, 14.73146284284273674793716154956, 15.62221148944790197278290452327, 16.19567497812788635025777869862, 16.78712289678490094396650275151, 18.16413241821439338040898620489, 18.80191706879992331639526760923, 19.75668650694500785700659659259, 20.09874581432712513638363638397, 21.2114816970778637519584100958, 21.7457521856781083012027316562

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