Properties

Label 2-483-23.22-c2-0-38
Degree $2$
Conductor $483$
Sign $0.928 - 0.371i$
Analytic cond. $13.1607$
Root an. cond. $3.62778$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.65·2-s + 1.73·3-s + 9.34·4-s + 3.50i·5-s + 6.32·6-s + 2.64i·7-s + 19.5·8-s + 2.99·9-s + 12.8i·10-s − 4.35i·11-s + 16.1·12-s − 8.22·13-s + 9.66i·14-s + 6.06i·15-s + 33.9·16-s − 11.9i·17-s + ⋯
L(s)  = 1  + 1.82·2-s + 0.577·3-s + 2.33·4-s + 0.700i·5-s + 1.05·6-s + 0.377i·7-s + 2.43·8-s + 0.333·9-s + 1.28i·10-s − 0.395i·11-s + 1.34·12-s − 0.632·13-s + 0.690i·14-s + 0.404i·15-s + 2.11·16-s − 0.704i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.928 - 0.371i$
Analytic conductor: \(13.1607\)
Root analytic conductor: \(3.62778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1),\ 0.928 - 0.371i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.130872967\)
\(L(\frac12)\) \(\approx\) \(6.130872967\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
7 \( 1 - 2.64iT \)
23 \( 1 + (8.55 + 21.3i)T \)
good2 \( 1 - 3.65T + 4T^{2} \)
5 \( 1 - 3.50iT - 25T^{2} \)
11 \( 1 + 4.35iT - 121T^{2} \)
13 \( 1 + 8.22T + 169T^{2} \)
17 \( 1 + 11.9iT - 289T^{2} \)
19 \( 1 - 15.8iT - 361T^{2} \)
29 \( 1 + 0.402T + 841T^{2} \)
31 \( 1 + 32.4T + 961T^{2} \)
37 \( 1 + 10.0iT - 1.36e3T^{2} \)
41 \( 1 + 15.1T + 1.68e3T^{2} \)
43 \( 1 + 44.7iT - 1.84e3T^{2} \)
47 \( 1 - 64.3T + 2.20e3T^{2} \)
53 \( 1 - 40.9iT - 2.80e3T^{2} \)
59 \( 1 + 40.4T + 3.48e3T^{2} \)
61 \( 1 + 21.4iT - 3.72e3T^{2} \)
67 \( 1 + 86.0iT - 4.48e3T^{2} \)
71 \( 1 - 6.72T + 5.04e3T^{2} \)
73 \( 1 + 81.7T + 5.32e3T^{2} \)
79 \( 1 + 13.2iT - 6.24e3T^{2} \)
83 \( 1 - 156. iT - 6.88e3T^{2} \)
89 \( 1 + 124. iT - 7.92e3T^{2} \)
97 \( 1 - 127. iT - 9.40e3T^{2} \)
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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

−11.02454570911306069451246061688, −10.32654064573662842973251075396, −9.018012732055856431827235906271, −7.68842004876378411066111252724, −6.91676793207128841338703333735, −5.98703467496139064558272607936, −5.03558294108793604598451230661, −3.91811031910053172991891556862, −2.98958754874377028641552828356, −2.17384223452468947457532255379, 1.70603949248342470228776753482, 2.95445167299201522832274426459, 4.05980816732244733461918885040, 4.77999444382595370574393253706, 5.71070634744151249215572192115, 6.92109274697805855398309034221, 7.61715478334277312020665352872, 8.876880216605490279892899517052, 10.00348075333283970218646567064, 11.04290244877063066882942192401

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