↑

Properties

Label 2-69-23.22-c4-0-4
Degree $2$
Conductor $69$
Sign $0.816 - 0.576i$
Analytic cond. $7.13252$
Root an. cond. $2.67067$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.44·2-s − 5.19·3-s − 13.9·4-s − 11.4i·5-s + 7.49·6-s − 12.7i·7-s + 43.1·8-s + 27·9-s + 16.5i·10-s + 213. i·11-s + 72.3·12-s + 181.·13-s + 18.4i·14-s + 59.5i·15-s + 160.·16-s − 347. i·17-s + ⋯
L(s)  = 1  − 0.360·2-s − 0.577·3-s − 0.869·4-s − 0.458i·5-s + 0.208·6-s − 0.261i·7-s + 0.674·8-s + 0.333·9-s + 0.165i·10-s + 1.76i·11-s + 0.502·12-s + 1.07·13-s + 0.0942i·14-s + 0.264i·15-s + 0.626·16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.816 - 0.576i$
Analytic conductor: \(7.13252\)
Root analytic conductor: \(2.67067\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :2),\ 0.816 - 0.576i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.821172 + 0.260733i\)
\(L(\frac12)\) \(\approx\) \(0.821172 + 0.260733i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
23 \( 1 + (-432. + 305. i)T \)
good2 \( 1 + 1.44T + 16T^{2} \)
5 \( 1 + 11.4iT - 625T^{2} \)
7 \( 1 + 12.7iT - 2.40e3T^{2} \)
11 \( 1 - 213. iT - 1.46e4T^{2} \)
13 \( 1 - 181.T + 2.85e4T^{2} \)
17 \( 1 + 347. iT - 8.35e4T^{2} \)
19 \( 1 - 701. iT - 1.30e5T^{2} \)
29 \( 1 + 637.T + 7.07e5T^{2} \)
31 \( 1 - 888.T + 9.23e5T^{2} \)
37 \( 1 - 1.55e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.69e3T + 2.82e6T^{2} \)
43 \( 1 + 1.68e3iT - 3.41e6T^{2} \)
47 \( 1 + 565.T + 4.87e6T^{2} \)
53 \( 1 - 4.64e3iT - 7.89e6T^{2} \)
59 \( 1 - 4.53e3T + 1.21e7T^{2} \)
61 \( 1 - 963. iT - 1.38e7T^{2} \)
67 \( 1 - 5.47e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.13e3T + 2.54e7T^{2} \)
73 \( 1 - 7.63e3T + 2.83e7T^{2} \)
79 \( 1 - 1.06e4iT - 3.89e7T^{2} \)
83 \( 1 + 3.35e3iT - 4.74e7T^{2} \)
89 \( 1 - 937. iT - 6.27e7T^{2} \)
97 \( 1 + 1.59e4iT - 8.85e7T^{2} \)
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

−13.96776697486988996928920729725, −12.89731474041994204149118627519, −12.03968287451780320316467957507, −10.45399024746333392508969035574, −9.632311075683777178644477249161, −8.382681253509199792043883390338, −7.04095952036191248357370794273, −5.25449975094098501950224933977, −4.18087058808565489078193123727, −1.17849979628701344830020332625, 0.73881955367034211357903569773, 3.54109989863067110322089739741, 5.28412556395585094311146991454, 6.53088899438899371604697198198, 8.304565325328164098375901612492, 9.135383897154207216194226299035, 10.73926842887672249568318343475, 11.24994667004583789838737082360, 13.05028672459088999194584126087, 13.63397085268572955688002862385

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