↑

Properties

Label 2-23-23.22-c24-0-12
Degree $2$
Conductor $23$
Sign $-0.777 + 0.628i$
Analytic cond. $83.9424$
Root an. cond. $9.16201$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.77e3·2-s + 5.73e5·3-s − 9.09e6·4-s + 4.41e8i·5-s − 1.58e9·6-s + 1.06e10i·7-s + 7.17e10·8-s + 4.60e10·9-s − 1.22e12i·10-s + 6.87e11i·11-s − 5.21e12·12-s − 2.40e13·13-s − 2.95e13i·14-s + 2.53e14i·15-s − 4.62e13·16-s + 5.52e14i·17-s + ⋯
L(s)  = 1  − 0.676·2-s + 1.07·3-s − 0.541·4-s + 1.80i·5-s − 0.729·6-s + 0.769i·7-s + 1.04·8-s + 0.163·9-s − 1.22i·10-s + 0.219i·11-s − 0.584·12-s − 1.03·13-s − 0.520i·14-s + 1.95i·15-s − 0.164·16-s + 0.947i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.777 + 0.628i$
Analytic conductor: \(83.9424\)
Root analytic conductor: \(9.16201\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :12),\ -0.777 + 0.628i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(1.158529506\)
\(L(\frac12)\) \(\approx\) \(1.158529506\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (1.70e16 - 1.37e16i)T \)
good2 \( 1 + 2.77e3T + 1.67e7T^{2} \)
3 \( 1 - 5.73e5T + 2.82e11T^{2} \)
5 \( 1 - 4.41e8iT - 5.96e16T^{2} \)
7 \( 1 - 1.06e10iT - 1.91e20T^{2} \)
11 \( 1 - 6.87e11iT - 9.84e24T^{2} \)
13 \( 1 + 2.40e13T + 5.42e26T^{2} \)
17 \( 1 - 5.52e14iT - 3.39e29T^{2} \)
19 \( 1 - 4.03e15iT - 4.89e30T^{2} \)
29 \( 1 - 4.00e17T + 1.25e35T^{2} \)
31 \( 1 - 1.18e18T + 6.20e35T^{2} \)
37 \( 1 + 3.13e18iT - 4.33e37T^{2} \)
41 \( 1 + 1.02e19T + 5.09e38T^{2} \)
43 \( 1 - 4.72e19iT - 1.59e39T^{2} \)
47 \( 1 - 2.24e20T + 1.35e40T^{2} \)
53 \( 1 + 1.30e20iT - 2.41e41T^{2} \)
59 \( 1 + 4.67e20T + 3.16e42T^{2} \)
61 \( 1 - 2.54e20iT - 7.04e42T^{2} \)
67 \( 1 + 4.53e21iT - 6.69e43T^{2} \)
71 \( 1 + 2.31e22T + 2.69e44T^{2} \)
73 \( 1 - 2.45e22T + 5.24e44T^{2} \)
79 \( 1 + 1.72e22iT - 3.49e45T^{2} \)
83 \( 1 - 2.03e23iT - 1.14e46T^{2} \)
89 \( 1 + 4.36e23iT - 6.10e46T^{2} \)
97 \( 1 - 3.23e22iT - 4.81e47T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.91411061616587901835363850569, −12.10155510056448583981509366076, −10.38029643966150310219407409268, −9.726154162219768176803648227751, −8.313160839075684279274237344345, −7.56580035841583978166180602008, −5.98914091400212283456325720228, −3.93517285165819742077837408472, −2.79547236365206364034989399988, −1.88158605101324080522863404317, 0.36730481079359344699677359330, 0.884191607746809885068909811538, 2.47351424424635006302458211906, 4.28472691310697749912398688043, 4.96293217336730698280776871083, 7.44209445761965175184918138936, 8.498555587548470233087606650484, 9.067385640211273280468196149647, 10.05117043984222518860999000122, 12.09171036223283217837169307611

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