Properties

Label 2-23-1.1-c3-0-1
Degree $2$
Conductor $23$
Sign $1$
Analytic cond. $1.35704$
Root an. cond. $1.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0323·2-s + 6.42·3-s − 7.99·4-s + 14.1·5-s − 0.207·6-s − 14.0·7-s + 0.517·8-s + 14.2·9-s − 0.456·10-s − 55.5·11-s − 51.3·12-s − 18.3·13-s + 0.453·14-s + 90.5·15-s + 63.9·16-s + 10.0·17-s − 0.460·18-s + 161.·19-s − 112.·20-s − 89.9·21-s + 1.79·22-s − 23·23-s + 3.32·24-s + 73.8·25-s + 0.592·26-s − 81.9·27-s + 112.·28-s + ⋯
L(s)  = 1  − 0.0114·2-s + 1.23·3-s − 0.999·4-s + 1.26·5-s − 0.0141·6-s − 0.756·7-s + 0.0228·8-s + 0.527·9-s − 0.0144·10-s − 1.52·11-s − 1.23·12-s − 0.390·13-s + 0.00865·14-s + 1.55·15-s + 0.999·16-s + 0.143·17-s − 0.00603·18-s + 1.94·19-s − 1.26·20-s − 0.934·21-s + 0.0174·22-s − 0.208·23-s + 0.0282·24-s + 0.591·25-s + 0.00447·26-s − 0.584·27-s + 0.756·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $1$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.358868247\)
\(L(\frac12)\) \(\approx\) \(1.358868247\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + 23T \)
good2 \( 1 + 0.0323T + 8T^{2} \)
3 \( 1 - 6.42T + 27T^{2} \)
5 \( 1 - 14.1T + 125T^{2} \)
7 \( 1 + 14.0T + 343T^{2} \)
11 \( 1 + 55.5T + 1.33e3T^{2} \)
13 \( 1 + 18.3T + 2.19e3T^{2} \)
17 \( 1 - 10.0T + 4.91e3T^{2} \)
19 \( 1 - 161.T + 6.85e3T^{2} \)
29 \( 1 - 183.T + 2.43e4T^{2} \)
31 \( 1 + 144.T + 2.97e4T^{2} \)
37 \( 1 - 181.T + 5.06e4T^{2} \)
41 \( 1 - 77.7T + 6.89e4T^{2} \)
43 \( 1 - 315.T + 7.95e4T^{2} \)
47 \( 1 + 524.T + 1.03e5T^{2} \)
53 \( 1 - 73.7T + 1.48e5T^{2} \)
59 \( 1 + 132.T + 2.05e5T^{2} \)
61 \( 1 - 236.T + 2.26e5T^{2} \)
67 \( 1 - 493.T + 3.00e5T^{2} \)
71 \( 1 + 806.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 + 599.T + 4.93e5T^{2} \)
83 \( 1 + 642.T + 5.71e5T^{2} \)
89 \( 1 - 883.T + 7.04e5T^{2} \)
97 \( 1 + 71.2T + 9.12e5T^{2} \)
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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

−17.72153460638264383312124300750, −16.04124027867043835060848323760, −14.41498782640713008098686995786, −13.62661473599703854307623302935, −12.89837041988241862465913436245, −9.975268380294332225708056230877, −9.392320644431521348886090010424, −7.894732367116790374871707639492, −5.43462989525735258810252461931, −2.92873018397997914043710060405, 2.92873018397997914043710060405, 5.43462989525735258810252461931, 7.894732367116790374871707639492, 9.392320644431521348886090010424, 9.975268380294332225708056230877, 12.89837041988241862465913436245, 13.62661473599703854307623302935, 14.41498782640713008098686995786, 16.04124027867043835060848323760, 17.72153460638264383312124300750

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