↑

Properties

Label 2-3528-7.2-c1-0-48
Degree $2$
Conductor $3528$
Sign $-0.991 - 0.126i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
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 − 1.73i)5-s + (−2 − 3.46i)11-s + 2·13-s + (−3 − 5.19i)17-s + (−4 + 6.92i)19-s + (0.500 + 0.866i)25-s − 6·29-s + (−4 − 6.92i)31-s + (1 − 1.73i)37-s − 2·41-s − 4·43-s + (−4 + 6.92i)47-s + (3 + 5.19i)53-s − 7.99·55-s + (3 − 5.19i)61-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (−0.603 − 1.04i)11-s + 0.554·13-s + (−0.727 − 1.26i)17-s + (−0.917 + 1.58i)19-s + (0.100 + 0.173i)25-s − 1.11·29-s + (−0.718 − 1.24i)31-s + (0.164 − 0.284i)37-s − 0.312·41-s − 0.609·43-s + (−0.583 + 1.01i)47-s + (0.412 + 0.713i)53-s − 1.07·55-s + (0.384 − 0.665i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5010744197\)
\(L(\frac12)\) \(\approx\) \(0.5010744197\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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

−8.242792853811672089924177324000, −7.62135792691190498769032676722, −6.57266779395104694305806105163, −5.75636346312353936597940121682, −5.36124022439734986559795617703, −4.31809853736027382662252033508, −3.51889863051200712724345189545, −2.41984590654836277065955255392, −1.42328680425892955404019516207, −0.13794029871789830177843569088, 1.77459012431860518593929449632, 2.40472872855139262739252710613, 3.44222563005466661107446109977, 4.38682891005626220835059926523, 5.15880243154139159184087563759, 6.10721642595436179116271076953, 6.83686683140558046296564218602, 7.18966060393141447416276885459, 8.407017069926959385513442878512, 8.788359837733482487556373952670

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