↑

Properties

Label 2-3328-104.35-c0-0-5
Degree $2$
Conductor $3328$
Sign $0.494 + 0.869i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
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  + (0.5 − 0.866i)3-s + (0.866 − 0.5i)7-s + (−0.5 + 0.866i)11-s − i·13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 0.999i·21-s + (0.866 + 0.5i)23-s + 25-s + 27-s + (−0.866 − 0.5i)29-s + (0.499 + 0.866i)33-s + (−0.866 − 0.5i)37-s + (−0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.866 − 0.5i)7-s + (−0.5 + 0.866i)11-s − i·13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 0.999i·21-s + (0.866 + 0.5i)23-s + 25-s + 27-s + (−0.866 − 0.5i)29-s + (0.499 + 0.866i)33-s + (−0.866 − 0.5i)37-s + (−0.866 − 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.494 + 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.675205350\)
\(L(\frac12)\) \(\approx\) \(1.675205350\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + iT \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.426758660376409448038504267131, −7.81893852638909113134461482204, −7.30626631837046261650350593796, −6.85728777652446313385837159738, −5.42894136127892640131958681388, −5.03831747369654903274961590876, −3.98328643826087084830161046515, −2.86866407088323778383609301520, −2.05767335750920249758629134482, −1.09881127569393785392347911417, 1.42684198494762525320702733547, 2.66367518451896418941454891437, 3.36379497060853414811939775600, 4.46972580171530219670621932100, 4.83414763811159628154728867745, 5.82902761938083594119122846226, 6.70403709228367105108970411166, 7.53916531997752403773992760001, 8.584038871376662075645837099073, 8.856610737209400278088929199033

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