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Properties

Label 2-3328-104.3-c0-0-0
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

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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: $\chi_{3328} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.494 - 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6156274991\)
\(L(\frac12)\) \(\approx\) \(0.6156274991\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.904183249766685046765697945933, −8.109343518296588089365405638909, −7.07708606479825755468804339791, −6.74702258678150738294652579708, −6.27524226927953594307164555778, −5.26840936172909663645000125092, −4.06336179750842552495307186203, −3.68685505991545581796567611172, −2.09550565646107279006192971878, −1.39208863363719946902491557056, 0.39421332947936960694720045424, 2.31866171898449589771157220623, 3.20281294872438892327869298487, 4.03722299392370999651503590502, 4.92031513197076334929695824721, 5.62878370638920069348868262498, 6.26497133662142034731152941151, 7.06982545044232633913440866421, 8.036667284539020177269289728746, 8.969777432898600095375406174886

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