Properties

Label 1-916-916.747-r1-0-0
Degree $1$
Conductor $916$
Sign $-0.995 - 0.0954i$
Analytic cond. $98.4378$
Root an. cond. $98.4378$
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  + (0.986 − 0.164i)3-s + (−0.879 − 0.475i)5-s + (0.401 − 0.915i)7-s + (0.945 − 0.324i)9-s + (0.0825 − 0.996i)11-s + (−0.879 − 0.475i)13-s + (−0.945 − 0.324i)15-s + (−0.879 − 0.475i)17-s + (0.879 − 0.475i)19-s + (0.245 − 0.969i)21-s + (0.677 − 0.735i)23-s + (0.546 + 0.837i)25-s + (0.879 − 0.475i)27-s + (−0.401 + 0.915i)29-s + (0.0825 − 0.996i)31-s + ⋯
L(s)  = 1  + (0.986 − 0.164i)3-s + (−0.879 − 0.475i)5-s + (0.401 − 0.915i)7-s + (0.945 − 0.324i)9-s + (0.0825 − 0.996i)11-s + (−0.879 − 0.475i)13-s + (−0.945 − 0.324i)15-s + (−0.879 − 0.475i)17-s + (0.879 − 0.475i)19-s + (0.245 − 0.969i)21-s + (0.677 − 0.735i)23-s + (0.546 + 0.837i)25-s + (0.879 − 0.475i)27-s + (−0.401 + 0.915i)29-s + (0.0825 − 0.996i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $-0.995 - 0.0954i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ -0.995 - 0.0954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09405595847 - 1.966178211i\)
\(L(\frac12)\) \(\approx\) \(0.09405595847 - 1.966178211i\)
\(L(1)\) \(\approx\) \(1.097862881 - 0.6115271032i\)
\(L(1)\) \(\approx\) \(1.097862881 - 0.6115271032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.986 - 0.164i)T \)
5 \( 1 + (-0.879 - 0.475i)T \)
7 \( 1 + (0.401 - 0.915i)T \)
11 \( 1 + (0.0825 - 0.996i)T \)
13 \( 1 + (-0.879 - 0.475i)T \)
17 \( 1 + (-0.879 - 0.475i)T \)
19 \( 1 + (0.879 - 0.475i)T \)
23 \( 1 + (0.677 - 0.735i)T \)
29 \( 1 + (-0.401 + 0.915i)T \)
31 \( 1 + (0.0825 - 0.996i)T \)
37 \( 1 + (0.245 - 0.969i)T \)
41 \( 1 + (0.945 + 0.324i)T \)
43 \( 1 + (-0.245 + 0.969i)T \)
47 \( 1 + (-0.945 + 0.324i)T \)
53 \( 1 + (-0.986 + 0.164i)T \)
59 \( 1 + (-0.245 - 0.969i)T \)
61 \( 1 + (0.945 - 0.324i)T \)
67 \( 1 + (-0.945 + 0.324i)T \)
71 \( 1 + (0.0825 + 0.996i)T \)
73 \( 1 + (0.546 + 0.837i)T \)
79 \( 1 + (0.401 + 0.915i)T \)
83 \( 1 + (-0.245 - 0.969i)T \)
89 \( 1 + T \)
97 \( 1 + (0.546 - 0.837i)T \)
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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

−22.08543895076779907509622809064, −21.22347299509456351556373497278, −20.43404272511399230884634528208, −19.5862723357124312105077177574, −19.16238870105725990863339807627, −18.27872854005363881959614869956, −17.51749971394992126247267945525, −16.23619492867089010474819038537, −15.34653142813206928083414203275, −15.0404529839966180603507862514, −14.364715340261046231897985618164, −13.33028940885075027787743870738, −12.267472634373457642509484929170, −11.77228417802117746112596996197, −10.68027775969375575708679889190, −9.6672169431917851163951672823, −9.025706154116562579723643925461, −8.03911493114107378851612086038, −7.43369661352155348319906256071, −6.60513319592873535221540787993, −5.06337978777729783446067026156, −4.368135573571255435247203677758, −3.36013087590563109343406645758, −2.447889538337454438294650480726, −1.63931232519982377644676823294, 0.38041256108008667046325459893, 1.1509458341115307928555906836, 2.62024568289073248337126362145, 3.438366086559964252195384831526, 4.3654457099987691281617503304, 5.073978324476046273032746695263, 6.699631075424791578006034060645, 7.51552652153673548085476312903, 8.02183241711980570085516266718, 8.93743001402432828599550271070, 9.66143610208788079764094006359, 10.93651722915313124688611143849, 11.45963558965890191244274124305, 12.77180890360325789812934766723, 13.179264709142466924925315109947, 14.25205270510884981234048628829, 14.73047514628109191819331633174, 15.80175091173421481728909612875, 16.34735288259440882788636353414, 17.34879867671897490714888696218, 18.29317870600584493340097925007, 19.201528481661303824041576460942, 19.89226356927151186222851763370, 20.2780286067561442548123063299, 21.04770123769955709386729819400

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