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Properties

Label 2-93-93.47-c0-0-0
Degree $2$
Conductor $93$
Sign $0.817 + 0.575i$
Analytic cond. $0.0464130$
Root an. cond. $0.215436$
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  + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 1.53i)7-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)12-s + (−0.5 − 0.363i)13-s + (−0.809 − 0.587i)16-s + (−0.5 + 0.363i)19-s + (1.30 − 0.951i)21-s + 25-s + (0.309 − 0.951i)27-s + (1.30 + 0.951i)28-s + (−0.809 − 0.587i)31-s + 36-s − 1.61·37-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 1.53i)7-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)12-s + (−0.5 − 0.363i)13-s + (−0.809 − 0.587i)16-s + (−0.5 + 0.363i)19-s + (1.30 − 0.951i)21-s + 25-s + (0.309 − 0.951i)27-s + (1.30 + 0.951i)28-s + (−0.809 − 0.587i)31-s + 36-s − 1.61·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(0.0464130\)
Root analytic conductor: \(0.215436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :0),\ 0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4619018582\)
\(L(\frac12)\) \(\approx\) \(0.4619018582\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.29554481691611927434390427375, −12.82898227684058972867745113271, −12.14890250289572350156878735868, −11.07675431074072360555066141752, −10.03532261497939277822132745853, −8.771530581454348144572623938258, −7.05804596571881808973016571167, −5.95144901391539568473880923115, −5.22099923391254890010555720139, −2.26669821585936102709468412872, 3.52379163668022342736277136955, 4.63388359943371166320155636122, 6.60718084941818390355171763636, 7.34947433847167642766250330443, 9.045382814740218732932819683367, 10.36797757186280028933745907370, 11.08431803071120187695208329795, 12.29045956804221355094943948134, 13.11029899315600657378924914600, 14.39792191631534296286288651217

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