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Properties

Label 2-92-23.22-c2-0-3
Degree $2$
Conductor $92$
Sign $0.861 + 0.508i$
Analytic cond. $2.50681$
Root an. cond. $1.58329$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.70·3-s − 7.32i·5-s − 2.18i·7-s + 4.70·9-s + 17.5i·11-s + 3.70·13-s − 27.1i·15-s + 16.8i·17-s − 9.50i·19-s − 8.08i·21-s + (19.8 + 11.6i)23-s − 28.6·25-s − 15.9·27-s − 41.9·29-s + 17.3·31-s + ⋯
L(s)  = 1  + 1.23·3-s − 1.46i·5-s − 0.312i·7-s + 0.522·9-s + 1.59i·11-s + 0.284·13-s − 1.80i·15-s + 0.989i·17-s − 0.500i·19-s − 0.385i·21-s + (0.861 + 0.508i)23-s − 1.14·25-s − 0.589·27-s − 1.44·29-s + 0.558·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.861 + 0.508i$
Analytic conductor: \(2.50681\)
Root analytic conductor: \(1.58329\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 92,\ (\ :1),\ 0.861 + 0.508i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.74230 - 0.475905i\)
\(L(\frac12)\) \(\approx\) \(1.74230 - 0.475905i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (-19.8 - 11.6i)T \)
good3 \( 1 - 3.70T + 9T^{2} \)
5 \( 1 + 7.32iT - 25T^{2} \)
7 \( 1 + 2.18iT - 49T^{2} \)
11 \( 1 - 17.5iT - 121T^{2} \)
13 \( 1 - 3.70T + 169T^{2} \)
17 \( 1 - 16.8iT - 289T^{2} \)
19 \( 1 + 9.50iT - 361T^{2} \)
29 \( 1 + 41.9T + 841T^{2} \)
31 \( 1 - 17.3T + 961T^{2} \)
37 \( 1 - 42.5iT - 1.36e3T^{2} \)
41 \( 1 + 71.5T + 1.68e3T^{2} \)
43 \( 1 + 78.3iT - 1.84e3T^{2} \)
47 \( 1 - 14.0T + 2.20e3T^{2} \)
53 \( 1 + 73.9iT - 2.80e3T^{2} \)
59 \( 1 - 53.2T + 3.48e3T^{2} \)
61 \( 1 - 6.66iT - 3.72e3T^{2} \)
67 \( 1 - 39.6iT - 4.48e3T^{2} \)
71 \( 1 + 61.1T + 5.04e3T^{2} \)
73 \( 1 - 24.4T + 5.32e3T^{2} \)
79 \( 1 + 130. iT - 6.24e3T^{2} \)
83 \( 1 - 29.1iT - 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + 128. iT - 9.40e3T^{2} \)
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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

−13.46784225942597093821101352740, −13.01062000932650139699077896764, −11.87783122262707137402149338725, −10.10133430889613242219085500854, −9.096472166990308192024914777641, −8.375884781851853625127972145274, −7.19829347370968169542722797566, −5.11723202207384691033456677587, −3.86392274702957005671351318979, −1.80265097587576663232961815849, 2.68817782145722512915924981544, 3.48007113302029812312150878153, 5.90187642470803943119405889932, 7.23028370215493107420326633587, 8.370222474401046197989788501290, 9.356560117379087953822222675722, 10.72845471440498464113368786581, 11.52963675376736026750031251813, 13.30463406033349681698727562506, 14.08280797924191046857877418985

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