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Properties

Label 2-23-23.22-c24-0-16
Degree $2$
Conductor $23$
Sign $0.242 - 0.970i$
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  − 1.21e3·2-s + 1.42e5·3-s − 1.53e7·4-s + 1.97e8i·5-s − 1.73e8·6-s + 6.84e9i·7-s + 3.89e10·8-s − 2.62e11·9-s − 2.40e11i·10-s − 1.27e12i·11-s − 2.18e12·12-s + 2.21e13·13-s − 8.31e12i·14-s + 2.82e13i·15-s + 2.09e14·16-s + 4.78e13i·17-s + ⋯
L(s)  = 1  − 0.296·2-s + 0.268·3-s − 0.912·4-s + 0.810i·5-s − 0.0796·6-s + 0.494i·7-s + 0.566·8-s − 0.927·9-s − 0.240i·10-s − 0.405i·11-s − 0.245·12-s + 0.948·13-s − 0.146i·14-s + 0.217i·15-s + 0.744·16-s + 0.0821i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.242 - 0.970i$
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: \((2,\ 23,\ (\ :12),\ 0.242 - 0.970i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(1.259269240\)
\(L(\frac12)\) \(\approx\) \(1.259269240\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-5.32e15 + 2.12e16i)T \)
good2 \( 1 + 1.21e3T + 1.67e7T^{2} \)
3 \( 1 - 1.42e5T + 2.82e11T^{2} \)
5 \( 1 - 1.97e8iT - 5.96e16T^{2} \)
7 \( 1 - 6.84e9iT - 1.91e20T^{2} \)
11 \( 1 + 1.27e12iT - 9.84e24T^{2} \)
13 \( 1 - 2.21e13T + 5.42e26T^{2} \)
17 \( 1 - 4.78e13iT - 3.39e29T^{2} \)
19 \( 1 + 1.36e15iT - 4.89e30T^{2} \)
29 \( 1 - 5.15e16T + 1.25e35T^{2} \)
31 \( 1 - 2.06e16T + 6.20e35T^{2} \)
37 \( 1 - 9.91e18iT - 4.33e37T^{2} \)
41 \( 1 - 1.69e19T + 5.09e38T^{2} \)
43 \( 1 + 6.15e19iT - 1.59e39T^{2} \)
47 \( 1 + 2.09e20T + 1.35e40T^{2} \)
53 \( 1 + 8.75e20iT - 2.41e41T^{2} \)
59 \( 1 - 1.74e21T + 3.16e42T^{2} \)
61 \( 1 - 4.45e21iT - 7.04e42T^{2} \)
67 \( 1 - 3.08e21iT - 6.69e43T^{2} \)
71 \( 1 + 1.51e22T + 2.69e44T^{2} \)
73 \( 1 - 3.28e22T + 5.24e44T^{2} \)
79 \( 1 - 5.89e22iT - 3.49e45T^{2} \)
83 \( 1 - 7.31e22iT - 1.14e46T^{2} \)
89 \( 1 - 1.90e23iT - 6.10e46T^{2} \)
97 \( 1 - 7.53e23iT - 4.81e47T^{2} \)
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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

−13.15568029278924862689390187743, −11.49003819104148276750782860830, −10.37204169309626450851533818502, −8.912968667032176625457210071565, −8.282903332405813919467326639785, −6.53819221995981889731381354078, −5.23082477268361841302868624715, −3.63685725700962329680195220107, −2.56768866950692731950051990161, −0.804241287490807359313994154611, 0.46581585683357340977713310996, 1.48796953742934968735615801080, 3.41962685125053666150945267930, 4.56908274411024733657963791461, 5.77933005044937422869813160694, 7.74205947950674599734489907515, 8.702204182663090679814559972776, 9.573006275621762551727600420757, 11.03717996290447542463532772888, 12.63021973241389318917362090020

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