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Properties

Label 1-92-92.55-r1-0-0
Degree $1$
Conductor $92$
Sign $-0.259 + 0.965i$
Analytic cond. $9.88677$
Root an. cond. $9.88677$
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  + (0.142 − 0.989i)3-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (0.142 + 0.989i)15-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.654 + 0.755i)33-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (0.142 + 0.989i)15-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.654 + 0.755i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $-0.259 + 0.965i$
Analytic conductor: \(9.88677\)
Root analytic conductor: \(9.88677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (1:\ ),\ -0.259 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3609339013 + 0.4707334798i\)
\(L(\frac12)\) \(\approx\) \(0.3609339013 + 0.4707334798i\)
\(L(1)\) \(\approx\) \(0.7654584523 + 0.009442246281i\)
\(L(1)\) \(\approx\) \(0.7654584523 + 0.009442246281i\)

Euler product

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

−29.90681644371988076110761214678, −28.41037895222802969396750367147, −27.53837454025864757848949012404, −26.8739642162069532490108583533, −25.89139786912172042344138917453, −24.4365642392819295717847253390, −23.30676434880302795289457371726, −22.52976024596552945675218687658, −20.99009681264095575212113345030, −20.394547154376053219516895406145, −19.43461678260867554181970081597, −17.743178415287845299040243054570, −16.69947036614961466434156567500, −15.550933665251795263249011125297, −14.877366698364558381825355043088, −13.45291211141360503767616409127, −11.900214757039425289997454785324, −10.837127487392737825638686859499, −9.81127882447923184281367410825, −8.27862305319517535542387475323, −7.41266034748705308573740773115, −5.09218855472529185117519465517, −4.38662591635681580779828266701, −2.88180005706009226167018485267, −0.25717312839695553691736600925, 1.82433457638488083222961275490, 3.29899585809516379239915982403, 5.18879428516421661926636101851, 6.665569798429187806374124009445, 7.94801088131005125513666019733, 8.59035756311435272419113939475, 10.69246598513412153888661645156, 11.89472486214394319899805281010, 12.55957908020031139731707733608, 14.173007515317698068376705419900, 14.97667585273362765150638654586, 16.358972487733017826125654373884, 17.77339558328210570413680342685, 18.86232345193947901994829125656, 19.302222185883445584007332867895, 20.77303233043927934422898637773, 21.97002788151570586704293931793, 23.48108409008183249659995753269, 23.900834409428376290238841041529, 25.00790679573916523042076059286, 26.19106547784125868185677283558, 27.25654418344104116402239870065, 28.42281370744912633195979789646, 29.42573934534099273200849215465, 30.60785419349033517523922898618

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