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Properties

Label 2-23-23.22-c12-0-9
Degree $2$
Conductor $23$
Sign $0.310 + 0.950i$
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  − 108.·2-s − 822.·3-s + 7.69e3·4-s − 2.21e4i·5-s + 8.93e4·6-s − 3.75e4i·7-s − 3.90e5·8-s + 1.45e5·9-s + 2.40e6i·10-s + 1.71e6i·11-s − 6.32e6·12-s + 2.82e6·13-s + 4.07e6i·14-s + 1.81e7i·15-s + 1.08e7·16-s + 3.09e7i·17-s + ⋯
L(s)  = 1  − 1.69·2-s − 1.12·3-s + 1.87·4-s − 1.41i·5-s + 1.91·6-s − 0.319i·7-s − 1.48·8-s + 0.273·9-s + 2.40i·10-s + 0.966i·11-s − 2.11·12-s + 0.585·13-s + 0.541i·14-s + 1.59i·15-s + 0.649·16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.310 + 0.950i$
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: \((2,\ 23,\ (\ :6),\ 0.310 + 0.950i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.352112 - 0.255490i\)
\(L(\frac12)\) \(\approx\) \(0.352112 - 0.255490i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-4.59e7 - 1.40e8i)T \)
good2 \( 1 + 108.T + 4.09e3T^{2} \)
3 \( 1 + 822.T + 5.31e5T^{2} \)
5 \( 1 + 2.21e4iT - 2.44e8T^{2} \)
7 \( 1 + 3.75e4iT - 1.38e10T^{2} \)
11 \( 1 - 1.71e6iT - 3.13e12T^{2} \)
13 \( 1 - 2.82e6T + 2.32e13T^{2} \)
17 \( 1 - 3.09e7iT - 5.82e14T^{2} \)
19 \( 1 + 6.04e7iT - 2.21e15T^{2} \)
29 \( 1 - 8.75e8T + 3.53e17T^{2} \)
31 \( 1 + 3.83e8T + 7.87e17T^{2} \)
37 \( 1 - 1.22e9iT - 6.58e18T^{2} \)
41 \( 1 - 2.42e9T + 2.25e19T^{2} \)
43 \( 1 - 2.41e9iT - 3.99e19T^{2} \)
47 \( 1 + 1.92e10T + 1.16e20T^{2} \)
53 \( 1 + 5.55e9iT - 4.91e20T^{2} \)
59 \( 1 - 2.95e10T + 1.77e21T^{2} \)
61 \( 1 + 1.00e11iT - 2.65e21T^{2} \)
67 \( 1 + 6.18e10iT - 8.18e21T^{2} \)
71 \( 1 + 7.65e10T + 1.64e22T^{2} \)
73 \( 1 - 1.34e11T + 2.29e22T^{2} \)
79 \( 1 + 4.58e11iT - 5.90e22T^{2} \)
83 \( 1 - 2.07e11iT - 1.06e23T^{2} \)
89 \( 1 - 1.66e11iT - 2.46e23T^{2} \)
97 \( 1 - 5.52e11iT - 6.93e23T^{2} \)
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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

−15.60353196314385869752878496800, −12.95117509211945657153290703870, −11.71444478906307754186198000811, −10.59585141324668627329574071362, −9.301098866110839081750707249176, −8.170453638668408155148301674504, −6.56164725991389657053198185568, −4.85724392060903855769642535350, −1.48869993801919791098846870338, −0.52320699989205849395545174593, 0.76956534811572905085793514792, 2.77346341691538464233069920995, 5.97934431345662166375596486318, 6.96755490085820669457548285750, 8.521406470585946508116053075012, 10.20177645442256334019964655787, 10.98235505737396982973781597935, 11.78519037472205276495173621915, 14.27622878157063827721387548259, 15.95591770053225912938148346615

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