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Properties

Label 2-23-23.22-c24-0-0
Degree $2$
Conductor $23$
Sign $-0.617 + 0.786i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.41e3·2-s − 6.00e5·3-s + 2.67e6·4-s + 2.08e8i·5-s + 2.64e9·6-s + 9.90e8i·7-s + 6.21e10·8-s + 7.76e10·9-s − 9.20e11i·10-s − 3.04e12i·11-s − 1.60e12·12-s − 2.90e13·13-s − 4.37e12i·14-s − 1.25e14i·15-s − 3.19e14·16-s + 7.59e14i·17-s + ⋯
L(s)  = 1  − 1.07·2-s − 1.12·3-s + 0.159·4-s + 0.854i·5-s + 1.21·6-s + 0.0715i·7-s + 0.904·8-s + 0.275·9-s − 0.920i·10-s − 0.970i·11-s − 0.180·12-s − 1.24·13-s − 0.0770i·14-s − 0.965i·15-s − 1.13·16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.617 + 0.786i$
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: \((2,\ 23,\ (\ :12),\ -0.617 + 0.786i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.04483929964\)
\(L(\frac12)\) \(\approx\) \(0.04483929964\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (1.35e16 - 1.72e16i)T \)
good2 \( 1 + 4.41e3T + 1.67e7T^{2} \)
3 \( 1 + 6.00e5T + 2.82e11T^{2} \)
5 \( 1 - 2.08e8iT - 5.96e16T^{2} \)
7 \( 1 - 9.90e8iT - 1.91e20T^{2} \)
11 \( 1 + 3.04e12iT - 9.84e24T^{2} \)
13 \( 1 + 2.90e13T + 5.42e26T^{2} \)
17 \( 1 - 7.59e14iT - 3.39e29T^{2} \)
19 \( 1 + 3.40e15iT - 4.89e30T^{2} \)
29 \( 1 + 1.97e17T + 1.25e35T^{2} \)
31 \( 1 - 1.05e18T + 6.20e35T^{2} \)
37 \( 1 - 5.88e18iT - 4.33e37T^{2} \)
41 \( 1 - 1.01e19T + 5.09e38T^{2} \)
43 \( 1 - 5.06e19iT - 1.59e39T^{2} \)
47 \( 1 + 2.15e20T + 1.35e40T^{2} \)
53 \( 1 - 6.98e20iT - 2.41e41T^{2} \)
59 \( 1 - 1.97e21T + 3.16e42T^{2} \)
61 \( 1 - 2.54e21iT - 7.04e42T^{2} \)
67 \( 1 - 1.45e22iT - 6.69e43T^{2} \)
71 \( 1 + 8.48e21T + 2.69e44T^{2} \)
73 \( 1 + 2.09e22T + 5.24e44T^{2} \)
79 \( 1 + 6.62e22iT - 3.49e45T^{2} \)
83 \( 1 + 1.10e23iT - 1.14e46T^{2} \)
89 \( 1 - 3.42e23iT - 6.10e46T^{2} \)
97 \( 1 - 2.21e23iT - 4.81e47T^{2} \)
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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

−13.34636753056107821548039221720, −11.66859522615858500180743897012, −10.83859255038974321767358735237, −9.925740322390150841112350032073, −8.491749123923919330435311032750, −7.16224971087185727658714847547, −6.02153812406725668162845159523, −4.63794815790636935840140257275, −2.75752501399244180292299482260, −1.07848599982911592406323343204, 0.03278029489108261243316201472, 0.68010755883884994502935465954, 2.01518748202478857084436592603, 4.50399948639640533312847699127, 5.26627723490276676578451808926, 6.96713948296805598033988526961, 8.160842871671478734250277637480, 9.538122628984783838288120052560, 10.30215329295780501600223581914, 11.81849531128731580026857561014

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