↑

Properties

Label 2-253-11.10-c2-0-15
Degree $2$
Conductor $253$
Sign $-0.989 - 0.147i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.88i·2-s + 1.17·3-s − 4.31·4-s + 1.65·5-s + 3.37i·6-s + 6.12i·7-s − 0.919i·8-s − 7.62·9-s + 4.77i·10-s + (10.8 + 1.62i)11-s − 5.05·12-s − 3.05i·13-s − 17.6·14-s + 1.94·15-s − 14.6·16-s + 25.1i·17-s + ⋯
L(s)  = 1  + 1.44i·2-s + 0.390·3-s − 1.07·4-s + 0.331·5-s + 0.562i·6-s + 0.874i·7-s − 0.114i·8-s − 0.847·9-s + 0.477i·10-s + (0.989 + 0.147i)11-s − 0.421·12-s − 0.235i·13-s − 1.26·14-s + 0.129·15-s − 0.913·16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.989 - 0.147i$
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: \((2,\ 253,\ (\ :1),\ -0.989 - 0.147i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.122267 + 1.64982i\)
\(L(\frac12)\) \(\approx\) \(0.122267 + 1.64982i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-10.8 - 1.62i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 - 2.88iT - 4T^{2} \)
3 \( 1 - 1.17T + 9T^{2} \)
5 \( 1 - 1.65T + 25T^{2} \)
7 \( 1 - 6.12iT - 49T^{2} \)
13 \( 1 + 3.05iT - 169T^{2} \)
17 \( 1 - 25.1iT - 289T^{2} \)
19 \( 1 - 4.90iT - 361T^{2} \)
29 \( 1 + 4.85iT - 841T^{2} \)
31 \( 1 - 31.3T + 961T^{2} \)
37 \( 1 - 18.5T + 1.36e3T^{2} \)
41 \( 1 + 17.6iT - 1.68e3T^{2} \)
43 \( 1 + 55.4iT - 1.84e3T^{2} \)
47 \( 1 - 77.9T + 2.20e3T^{2} \)
53 \( 1 - 47.3T + 2.80e3T^{2} \)
59 \( 1 - 25.3T + 3.48e3T^{2} \)
61 \( 1 - 72.9iT - 3.72e3T^{2} \)
67 \( 1 + 18.1T + 4.48e3T^{2} \)
71 \( 1 + 68.4T + 5.04e3T^{2} \)
73 \( 1 - 74.7iT - 5.32e3T^{2} \)
79 \( 1 - 128. iT - 6.24e3T^{2} \)
83 \( 1 + 33.0iT - 6.88e3T^{2} \)
89 \( 1 + 76.4T + 7.92e3T^{2} \)
97 \( 1 - 184.T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.33562098919540558613853339841, −11.46375967891621229027690402513, −10.02176103926331839584779712570, −8.757914965283814186563617650852, −8.528249809476816769628968194164, −7.30252371625405133177690544425, −5.96495978897702530284308216997, −5.76559079094049159308670628592, −4.05432406145390488576361639005, −2.27360385276600012165241882548, 0.840097344599400043962509820580, 2.39704220322142049714223167345, 3.49238402094828097669813968098, 4.58188047957195671796731306361, 6.27401877532939552294129001036, 7.50705061895305660493285217416, 8.952159605823081972978402754216, 9.557445021013173631521825850709, 10.49080515625365259574494489376, 11.55983326025692146190539578567

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