↑

Properties

Label 2-2717-2717.1429-c0-0-0
Degree $2$
Conductor $2717$
Sign $0.806 + 0.591i$
Analytic cond. $1.35595$
Root an. cond. $1.16445$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.454 − 0.165i)2-s + (−0.346 − 1.96i)3-s + (−0.586 − 0.492i)4-s + (−0.167 + 0.950i)6-s + (−0.374 + 0.648i)7-s + (0.427 + 0.739i)8-s + (−2.80 + 1.01i)9-s + (0.5 + 0.866i)11-s + (−0.764 + 1.32i)12-s + (−0.173 + 0.984i)13-s + (0.277 − 0.233i)14-s + (0.0612 + 0.347i)16-s + 1.44·18-s + (−0.438 − 0.898i)19-s + (1.40 + 0.511i)21-s + (−0.0840 − 0.476i)22-s + ⋯
L(s)  = 1  + (−0.454 − 0.165i)2-s + (−0.346 − 1.96i)3-s + (−0.586 − 0.492i)4-s + (−0.167 + 0.950i)6-s + (−0.374 + 0.648i)7-s + (0.427 + 0.739i)8-s + (−2.80 + 1.01i)9-s + (0.5 + 0.866i)11-s + (−0.764 + 1.32i)12-s + (−0.173 + 0.984i)13-s + (0.277 − 0.233i)14-s + (0.0612 + 0.347i)16-s + 1.44·18-s + (−0.438 − 0.898i)19-s + (1.40 + 0.511i)21-s + (−0.0840 − 0.476i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2717\)    =    \(11 \cdot 13 \cdot 19\)
Sign: $0.806 + 0.591i$
Analytic conductor: \(1.35595\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2717} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2717,\ (\ :0),\ 0.806 + 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5012200857\)
\(L(\frac12)\) \(\approx\) \(0.5012200857\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.438 + 0.898i)T \)
good2 \( 1 + (0.454 + 0.165i)T + (0.766 + 0.642i)T^{2} \)
3 \( 1 + (0.346 + 1.96i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.374 - 0.648i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.0534 - 0.0448i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.213 - 1.21i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.856 - 0.718i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.848 + 1.46i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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

−8.928199083382149478501652263091, −8.202776268317313223892159856364, −7.38661469507117722319050297433, −6.61327761655809672545798243186, −6.19054313571055904685518420410, −5.24709141357581972499424081156, −4.40405887719544824180994774189, −2.61249367796948238821189118904, −2.01782604081966149937157347228, −1.02974141158706146493604301360, 0.49578995086526807509890301990, 3.08279526297958980866602926486, 3.68612722877817369030602721808, 4.12975793999769132857785119071, 5.17221357849990826170892452503, 5.74736129601734474784136026961, 6.77407785624348404453264288126, 7.932574801070789870499977312783, 8.560476631770979592767267451965, 9.202112376306712815126115846074

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