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Properties

Label 2-253-11.10-c2-0-19
Degree $2$
Conductor $253$
Sign $0.474 - 0.880i$
Analytic cond. $6.89375$
Root an. cond. $2.62559$
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  + 1.09i·2-s + 1.62·3-s + 2.80·4-s + 7.64·5-s + 1.77i·6-s + 6.41i·7-s + 7.43i·8-s − 6.36·9-s + 8.34i·10-s + (−5.22 + 9.68i)11-s + 4.55·12-s − 11.7i·13-s − 7.00·14-s + 12.4·15-s + 3.11·16-s − 28.2i·17-s + ⋯
L(s)  = 1  + 0.546i·2-s + 0.541·3-s + 0.701·4-s + 1.52·5-s + 0.295i·6-s + 0.916i·7-s + 0.929i·8-s − 0.707·9-s + 0.834i·10-s + (−0.474 + 0.880i)11-s + 0.379·12-s − 0.901i·13-s − 0.500·14-s + 0.826·15-s + 0.194·16-s − 1.66i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.474 - 0.880i$
Analytic conductor: \(6.89375\)
Root analytic conductor: \(2.62559\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :1),\ 0.474 - 0.880i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.24903 + 1.34227i\)
\(L(\frac12)\) \(\approx\) \(2.24903 + 1.34227i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (5.22 - 9.68i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 - 1.09iT - 4T^{2} \)
3 \( 1 - 1.62T + 9T^{2} \)
5 \( 1 - 7.64T + 25T^{2} \)
7 \( 1 - 6.41iT - 49T^{2} \)
13 \( 1 + 11.7iT - 169T^{2} \)
17 \( 1 + 28.2iT - 289T^{2} \)
19 \( 1 - 0.740iT - 361T^{2} \)
29 \( 1 + 9.49iT - 841T^{2} \)
31 \( 1 + 46.8T + 961T^{2} \)
37 \( 1 - 67.1T + 1.36e3T^{2} \)
41 \( 1 + 10.1iT - 1.68e3T^{2} \)
43 \( 1 + 53.5iT - 1.84e3T^{2} \)
47 \( 1 - 10.1T + 2.20e3T^{2} \)
53 \( 1 + 4.65T + 2.80e3T^{2} \)
59 \( 1 - 56.0T + 3.48e3T^{2} \)
61 \( 1 + 13.8iT - 3.72e3T^{2} \)
67 \( 1 + 4.31T + 4.48e3T^{2} \)
71 \( 1 - 6.30T + 5.04e3T^{2} \)
73 \( 1 - 64.0iT - 5.32e3T^{2} \)
79 \( 1 + 31.1iT - 6.24e3T^{2} \)
83 \( 1 - 109. iT - 6.88e3T^{2} \)
89 \( 1 + 93.6T + 7.92e3T^{2} \)
97 \( 1 - 1.35T + 9.40e3T^{2} \)
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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

−12.03000773627543535661265836114, −10.96933937699219480158401724698, −9.816281228978784680112355442036, −9.120582770800307423301389215186, −8.005259919310189470662122089352, −6.98228938323976656558675877019, −5.64385959945664296830016348967, −5.42113783062708624333922860826, −2.70587109447620971747463137789, −2.23776107757652877939482700081, 1.52596548991726592444629850421, 2.60059619554456336775361151038, 3.83156293519026550435063725641, 5.76881454051929699303305776979, 6.43069806996408699942420229663, 7.76683727697879328865884533476, 8.955486768560773663828305990326, 9.890476063088827557006133600518, 10.72725544699549071110605043183, 11.33143811353773539900804399640

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