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Properties

Label 2-253-11.10-c2-0-18
Degree $2$
Conductor $253$
Sign $0.833 - 0.552i$
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  + 3.71i·2-s − 2.82·3-s − 9.78·4-s + 2.21·5-s − 10.4i·6-s − 5.12i·7-s − 21.4i·8-s − 1.04·9-s + 8.23i·10-s + (9.16 − 6.08i)11-s + 27.5·12-s + 2.92i·13-s + 19.0·14-s − 6.25·15-s + 40.5·16-s − 12.0i·17-s + ⋯
L(s)  = 1  + 1.85i·2-s − 0.940·3-s − 2.44·4-s + 0.443·5-s − 1.74i·6-s − 0.731i·7-s − 2.68i·8-s − 0.116·9-s + 0.823i·10-s + (0.833 − 0.552i)11-s + 2.29·12-s + 0.224i·13-s + 1.35·14-s − 0.417·15-s + 2.53·16-s − 0.708i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.833 - 0.552i$
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: \((2,\ 253,\ (\ :1),\ 0.833 - 0.552i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.728182 + 0.219612i\)
\(L(\frac12)\) \(\approx\) \(0.728182 + 0.219612i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-9.16 + 6.08i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 - 3.71iT - 4T^{2} \)
3 \( 1 + 2.82T + 9T^{2} \)
5 \( 1 - 2.21T + 25T^{2} \)
7 \( 1 + 5.12iT - 49T^{2} \)
13 \( 1 - 2.92iT - 169T^{2} \)
17 \( 1 + 12.0iT - 289T^{2} \)
19 \( 1 + 1.76iT - 361T^{2} \)
29 \( 1 + 7.43iT - 841T^{2} \)
31 \( 1 + 15.3T + 961T^{2} \)
37 \( 1 - 53.4T + 1.36e3T^{2} \)
41 \( 1 + 58.9iT - 1.68e3T^{2} \)
43 \( 1 + 45.8iT - 1.84e3T^{2} \)
47 \( 1 - 5.73T + 2.20e3T^{2} \)
53 \( 1 + 90.1T + 2.80e3T^{2} \)
59 \( 1 - 71.4T + 3.48e3T^{2} \)
61 \( 1 + 26.1iT - 3.72e3T^{2} \)
67 \( 1 - 20.7T + 4.48e3T^{2} \)
71 \( 1 - 20.2T + 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 + 13.9iT - 6.24e3T^{2} \)
83 \( 1 + 86.4iT - 6.88e3T^{2} \)
89 \( 1 + 29.8T + 7.92e3T^{2} \)
97 \( 1 + 79.0T + 9.40e3T^{2} \)
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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

−11.96321346127232001150304740696, −10.88917940532138367055390119261, −9.639037637462278258577804394840, −8.833078479661988056831250284462, −7.62585763703528833947305867014, −6.70277060956268660832530640327, −5.98853632248213511284610531111, −5.14872496818217818719591101682, −3.97596178774092732395001198902, −0.50462492750501281215532622621, 1.35177480376511975819286678930, 2.67508913921440626306147032429, 4.14401827217144789117721412147, 5.29647434536082133590784330809, 6.25905593412270191987115921409, 8.287724653510086834304429259561, 9.368023867878179122570752503227, 9.977299358383131329194031286735, 11.12507649102234221088493935611, 11.53326798508993595030767083158

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