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Properties

Label 2-483-23.22-c2-0-30
Degree $2$
Conductor $483$
Sign $0.743 - 0.669i$
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  + 3.29·2-s + 1.73·3-s + 6.87·4-s + 7.57i·5-s + 5.71·6-s − 2.64i·7-s + 9.47·8-s + 2.99·9-s + 24.9i·10-s + 5.86i·11-s + 11.9·12-s + 8.02·13-s − 8.72i·14-s + 13.1i·15-s + 3.74·16-s − 15.4i·17-s + ⋯
L(s)  = 1  + 1.64·2-s + 0.577·3-s + 1.71·4-s + 1.51i·5-s + 0.951·6-s − 0.377i·7-s + 1.18·8-s + 0.333·9-s + 2.49i·10-s + 0.533i·11-s + 0.992·12-s + 0.617·13-s − 0.623i·14-s + 0.874i·15-s + 0.234·16-s − 0.910i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.743 - 0.669i$
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: \((2,\ 483,\ (\ :1),\ 0.743 - 0.669i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.335217266\)
\(L(\frac12)\) \(\approx\) \(5.335217266\)
\(L(2)\) 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 \( 1 + (-15.3 - 17.0i)T \)
good2 \( 1 - 3.29T + 4T^{2} \)
5 \( 1 - 7.57iT - 25T^{2} \)
11 \( 1 - 5.86iT - 121T^{2} \)
13 \( 1 - 8.02T + 169T^{2} \)
17 \( 1 + 15.4iT - 289T^{2} \)
19 \( 1 + 2.95iT - 361T^{2} \)
29 \( 1 + 45.9T + 841T^{2} \)
31 \( 1 - 57.0T + 961T^{2} \)
37 \( 1 + 39.0iT - 1.36e3T^{2} \)
41 \( 1 - 19.9T + 1.68e3T^{2} \)
43 \( 1 + 65.5iT - 1.84e3T^{2} \)
47 \( 1 + 82.3T + 2.20e3T^{2} \)
53 \( 1 - 26.9iT - 2.80e3T^{2} \)
59 \( 1 - 44.4T + 3.48e3T^{2} \)
61 \( 1 + 93.3iT - 3.72e3T^{2} \)
67 \( 1 - 3.33iT - 4.48e3T^{2} \)
71 \( 1 + 62.7T + 5.04e3T^{2} \)
73 \( 1 - 49.4T + 5.32e3T^{2} \)
79 \( 1 + 3.58iT - 6.24e3T^{2} \)
83 \( 1 + 122. iT - 6.88e3T^{2} \)
89 \( 1 + 27.0iT - 7.92e3T^{2} \)
97 \( 1 + 44.3iT - 9.40e3T^{2} \)
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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

−11.17077189473896364215846203263, −10.28619822375202508819276947168, −9.233998561428698243070875424455, −7.60829757767592823258393827358, −7.01465694197560048673597964476, −6.22879392601262405007590676289, −5.02743617622963070133223415313, −3.83897029525734023543799539418, −3.16686631852434159514662134714, −2.18936635072667905523938834236, 1.43249009332640968943119157605, 2.90238583165702012442948326748, 4.01960975290765419277848793184, 4.78436531642322085465430479883, 5.71131806958575473778902975404, 6.53789698949283789446242087863, 8.132080178559534759176558281360, 8.641979216305893192059271389313, 9.700031397204694912669598717902, 11.09541637086352212894379514131

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