Properties

Label 1-920-920.293-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.277 - 0.960i$
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  + (0.540 − 0.841i)3-s + (0.281 − 0.959i)7-s + (−0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.281 + 0.959i)13-s + (0.989 − 0.142i)17-s + (0.142 − 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.989 − 0.142i)27-s + (−0.142 − 0.989i)29-s + (0.841 − 0.540i)31-s + (0.281 + 0.959i)33-s + (0.909 − 0.415i)37-s + (0.959 + 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (0.281 − 0.959i)7-s + (−0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (0.281 + 0.959i)13-s + (0.989 − 0.142i)17-s + (0.142 − 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.989 − 0.142i)27-s + (−0.142 − 0.989i)29-s + (0.841 − 0.540i)31-s + (0.281 + 0.959i)33-s + (0.909 − 0.415i)37-s + (0.959 + 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021448265 - 1.358339124i\)
\(L(\frac12)\) \(\approx\) \(1.021448265 - 1.358339124i\)
\(L(1)\) \(\approx\) \(1.138087224 - 0.5572506516i\)
\(L(1)\) \(\approx\) \(1.138087224 - 0.5572506516i\)

Euler product

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

−21.875834939445843608508849713108, −21.32144427819931580951091015716, −20.68742373704568654368912965738, −19.90816901438740883566914034716, −18.82344880503092999374035766830, −18.44018447102929351722722758479, −17.281784238837403219643795589467, −16.25501620773468401779789354548, −15.82250989846326927690103574456, −14.89630013628424742674060023796, −14.377388451477332936746276972803, −13.37954362964139200999174820347, −12.48617882582657011539400062323, −11.52781182695610214277831814941, −10.5779420969663222125125657264, −10.011353505912594490062058770887, −8.94935396371950271211265867364, −8.239233588201541852018042788068, −7.70878393913864361000775695623, −5.91971420038459356774488719331, −5.512188381826892147635715585415, −4.51372635118381421555519681615, −3.18318140786283490984532656123, −2.891536987078554724408554038101, −1.43425223687998678606709139660, 0.724136616344361822906672854919, 1.81329632449489497654258271301, 2.72820847025101067922367847019, 3.878404413339619861114547377999, 4.743650735314555290794408036268, 6.02041750205753853747802761519, 7.02151874914186275704779975097, 7.53577175637567669004909422293, 8.32570606868745258590579435420, 9.41495351006350384596864170895, 10.132044296565242485895013124894, 11.32257935402854034903397393479, 11.96168047487792959255925111082, 13.08378797854412714002964259084, 13.52526077340082381510628059936, 14.35408696863789781593298276282, 15.060573891311400388777163083319, 16.121117590526321115426180750163, 17.09179594646016246350146463630, 17.7456296246943516051119873410, 18.54711400369129247425773320496, 19.297193852939090181823420139742, 20.04337582127411727702064364564, 20.80482339512821105705507560880, 21.310765399857554759103817371012

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