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Properties

Label 2-23-23.22-c14-0-9
Degree $2$
Conductor $23$
Sign $-0.575 - 0.818i$
Analytic cond. $28.5956$
Root an. cond. $5.34749$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 137.·2-s + 2.03e3·3-s + 2.60e3·4-s + 5.83e4i·5-s + 2.80e5·6-s + 1.12e6i·7-s − 1.89e6·8-s − 6.33e5·9-s + 8.04e6i·10-s − 7.80e6i·11-s + 5.29e6·12-s − 7.61e7·13-s + 1.55e8i·14-s + 1.18e8i·15-s − 3.04e8·16-s + 4.29e8i·17-s + ⋯
L(s)  = 1  + 1.07·2-s + 0.931·3-s + 0.158·4-s + 0.747i·5-s + 1.00·6-s + 1.37i·7-s − 0.905·8-s − 0.132·9-s + 0.804i·10-s − 0.400i·11-s + 0.147·12-s − 1.21·13-s + 1.47i·14-s + 0.695i·15-s − 1.13·16-s + 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.575 - 0.818i$
Analytic conductor: \(28.5956\)
Root analytic conductor: \(5.34749\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :7),\ -0.575 - 0.818i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(2.832938655\)
\(L(\frac12)\) \(\approx\) \(2.832938655\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-1.95e9 - 2.78e9i)T \)
good2 \( 1 - 137.T + 1.63e4T^{2} \)
3 \( 1 - 2.03e3T + 4.78e6T^{2} \)
5 \( 1 - 5.83e4iT - 6.10e9T^{2} \)
7 \( 1 - 1.12e6iT - 6.78e11T^{2} \)
11 \( 1 + 7.80e6iT - 3.79e14T^{2} \)
13 \( 1 + 7.61e7T + 3.93e15T^{2} \)
17 \( 1 - 4.29e8iT - 1.68e17T^{2} \)
19 \( 1 + 8.27e8iT - 7.99e17T^{2} \)
29 \( 1 - 1.59e10T + 2.97e20T^{2} \)
31 \( 1 - 2.16e10T + 7.56e20T^{2} \)
37 \( 1 - 9.88e10iT - 9.01e21T^{2} \)
41 \( 1 - 1.77e10T + 3.79e22T^{2} \)
43 \( 1 - 4.10e11iT - 7.38e22T^{2} \)
47 \( 1 + 3.61e11T + 2.56e23T^{2} \)
53 \( 1 - 1.13e12iT - 1.37e24T^{2} \)
59 \( 1 - 3.54e12T + 6.19e24T^{2} \)
61 \( 1 + 5.08e12iT - 9.87e24T^{2} \)
67 \( 1 + 9.73e12iT - 3.67e25T^{2} \)
71 \( 1 + 4.25e12T + 8.27e25T^{2} \)
73 \( 1 - 6.04e12T + 1.22e26T^{2} \)
79 \( 1 - 2.57e13iT - 3.68e26T^{2} \)
83 \( 1 + 8.42e12iT - 7.36e26T^{2} \)
89 \( 1 + 9.49e11iT - 1.95e27T^{2} \)
97 \( 1 + 3.57e13iT - 6.52e27T^{2} \)
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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

−14.89631340112968946413880185482, −13.91663015481178630241861256879, −12.68550085517489373119014608619, −11.49191926124948221761197402586, −9.456128121592970838344263826400, −8.341032062610889350480592330837, −6.36842642661179325316214411957, −4.99277443641437968291931319540, −3.13622564688082881642949190541, −2.54341695529980864291094332238, 0.53300324642191864321864898085, 2.65772137203922887114558943010, 4.05424676179421165761315022147, 5.06655649381747242255668370046, 7.15110564623492990168954960028, 8.653672028500188480130129518505, 9.974685831932010504570459683236, 12.02928743550410436474240517558, 13.13780231546120866806520576745, 14.09958243796250274527215557763

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