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Properties

Label 2-253-11.10-c2-0-32
Degree $2$
Conductor $253$
Sign $0.999 + 0.0110i$
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  + 0.703i·2-s + 3.72·3-s + 3.50·4-s − 1.98·5-s + 2.62i·6-s − 11.6i·7-s + 5.28i·8-s + 4.88·9-s − 1.39i·10-s + (10.9 + 0.121i)11-s + 13.0·12-s − 9.72i·13-s + 8.16·14-s − 7.37·15-s + 10.3·16-s − 0.249i·17-s + ⋯
L(s)  = 1  + 0.351i·2-s + 1.24·3-s + 0.876·4-s − 0.396·5-s + 0.436i·6-s − 1.65i·7-s + 0.660i·8-s + 0.542·9-s − 0.139i·10-s + (0.999 + 0.0110i)11-s + 1.08·12-s − 0.748i·13-s + 0.583·14-s − 0.491·15-s + 0.644·16-s − 0.0146i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0110i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.999 + 0.0110i$
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: \((2,\ 253,\ (\ :1),\ 0.999 + 0.0110i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.70866 - 0.0149918i\)
\(L(\frac12)\) \(\approx\) \(2.70866 - 0.0149918i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-10.9 - 0.121i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 - 0.703iT - 4T^{2} \)
3 \( 1 - 3.72T + 9T^{2} \)
5 \( 1 + 1.98T + 25T^{2} \)
7 \( 1 + 11.6iT - 49T^{2} \)
13 \( 1 + 9.72iT - 169T^{2} \)
17 \( 1 + 0.249iT - 289T^{2} \)
19 \( 1 - 29.1iT - 361T^{2} \)
29 \( 1 - 32.3iT - 841T^{2} \)
31 \( 1 + 43.2T + 961T^{2} \)
37 \( 1 - 42.4T + 1.36e3T^{2} \)
41 \( 1 - 34.4iT - 1.68e3T^{2} \)
43 \( 1 + 32.0iT - 1.84e3T^{2} \)
47 \( 1 - 18.1T + 2.20e3T^{2} \)
53 \( 1 - 48.8T + 2.80e3T^{2} \)
59 \( 1 + 27.2T + 3.48e3T^{2} \)
61 \( 1 - 46.8iT - 3.72e3T^{2} \)
67 \( 1 + 35.9T + 4.48e3T^{2} \)
71 \( 1 + 63.5T + 5.04e3T^{2} \)
73 \( 1 - 59.6iT - 5.32e3T^{2} \)
79 \( 1 + 119. iT - 6.24e3T^{2} \)
83 \( 1 + 139. iT - 6.88e3T^{2} \)
89 \( 1 + 9.28T + 7.92e3T^{2} \)
97 \( 1 + 122.T + 9.40e3T^{2} \)
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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

−11.74755871224391693957054281670, −10.75752945736669425270171852756, −9.929673692842635354509119789758, −8.638391240202549915205303502739, −7.62485040845497734183078780732, −7.30313051418126321731244021062, −5.94045552045172362715829276027, −4.00813013116330645676038425597, −3.27770158339498103592516552427, −1.53867297230580754464225247129, 2.01229723973156330599118708762, 2.78430705112601100006960322928, 4.00646842570972475845643572420, 5.83533094222357167113014785460, 6.94612299575247556111803390198, 8.048377284781023731792356647376, 9.131947571385267081582417465955, 9.435248254541981103012650028581, 11.25366809795403470568580135469, 11.68255308463803330866394890195

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