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Properties

Label 2-23-23.22-c12-0-4
Degree $2$
Conductor $23$
Sign $-0.669 - 0.742i$
Analytic cond. $21.0218$
Root an. cond. $4.58495$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 111.·2-s + 792.·3-s + 8.38e3·4-s − 7.56e3i·5-s − 8.84e4·6-s + 2.31e5i·7-s − 4.79e5·8-s + 9.60e4·9-s + 8.45e5i·10-s − 9.56e5i·11-s + 6.64e6·12-s + 1.33e6·13-s − 2.59e7i·14-s − 5.99e6i·15-s + 1.91e7·16-s + 2.04e7i·17-s + ⋯
L(s)  = 1  − 1.74·2-s + 1.08·3-s + 2.04·4-s − 0.484i·5-s − 1.89·6-s + 1.97i·7-s − 1.82·8-s + 0.180·9-s + 0.845i·10-s − 0.539i·11-s + 2.22·12-s + 0.275·13-s − 3.44i·14-s − 0.526i·15-s + 1.14·16-s + 0.848i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(21.0218\)
Root analytic conductor: \(4.58495\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :6),\ -0.669 - 0.742i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.262143 + 0.589131i\)
\(L(\frac12)\) \(\approx\) \(0.262143 + 0.589131i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (9.91e7 + 1.09e8i)T \)
good2 \( 1 + 111.T + 4.09e3T^{2} \)
3 \( 1 - 792.T + 5.31e5T^{2} \)
5 \( 1 + 7.56e3iT - 2.44e8T^{2} \)
7 \( 1 - 2.31e5iT - 1.38e10T^{2} \)
11 \( 1 + 9.56e5iT - 3.13e12T^{2} \)
13 \( 1 - 1.33e6T + 2.32e13T^{2} \)
17 \( 1 - 2.04e7iT - 5.82e14T^{2} \)
19 \( 1 - 1.74e6iT - 2.21e15T^{2} \)
29 \( 1 + 5.12e8T + 3.53e17T^{2} \)
31 \( 1 + 1.11e9T + 7.87e17T^{2} \)
37 \( 1 - 2.11e9iT - 6.58e18T^{2} \)
41 \( 1 - 5.34e8T + 2.25e19T^{2} \)
43 \( 1 - 1.05e10iT - 3.99e19T^{2} \)
47 \( 1 - 2.06e8T + 1.16e20T^{2} \)
53 \( 1 - 3.93e10iT - 4.91e20T^{2} \)
59 \( 1 + 2.74e10T + 1.77e21T^{2} \)
61 \( 1 + 7.12e10iT - 2.65e21T^{2} \)
67 \( 1 - 1.12e11iT - 8.18e21T^{2} \)
71 \( 1 - 1.71e11T + 1.64e22T^{2} \)
73 \( 1 + 2.22e11T + 2.29e22T^{2} \)
79 \( 1 - 2.90e11iT - 5.90e22T^{2} \)
83 \( 1 - 1.27e11iT - 1.06e23T^{2} \)
89 \( 1 - 1.91e11iT - 2.46e23T^{2} \)
97 \( 1 + 9.67e11iT - 6.93e23T^{2} \)
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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

−15.67661679305325329054318290104, −14.69186155205338251981054367128, −12.62855229031380427241524820318, −11.21814252214629265133842139223, −9.442237138813515955966560777756, −8.706945197326683967328471546025, −8.122764517606512445178942928342, −5.99002767475535062486603256803, −2.81772379937839274681819070737, −1.69876342794540545954932173147, 0.33712559867235700213676435444, 1.87042725290391928831163772568, 3.56057857116075944448456172835, 7.11157947263613603901193632560, 7.65394929782067996085914792686, 9.131519686813720429811092484467, 10.18542210887034652416157278749, 11.16648018132562549076725156607, 13.55095814683993131494511384810, 14.65562301706829846826888908943

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