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Properties

Label 2-3328-13.11-c0-0-1
Degree $2$
Conductor $3328$
Sign $0.884 + 0.466i$
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  + (0.366 + 0.366i)5-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)13-s + (0.866 + 0.5i)17-s − 0.732i·25-s + (−0.5 − 0.866i)29-s + (1.86 + 0.5i)37-s + (0.133 − 0.5i)41-s + (0.5 − 0.133i)45-s + (0.866 − 0.5i)49-s − 1.73·53-s + (−0.866 + 1.5i)61-s + (0.133 − 0.5i)65-s + (0.366 − 0.366i)73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (0.366 + 0.366i)5-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)13-s + (0.866 + 0.5i)17-s − 0.732i·25-s + (−0.5 − 0.866i)29-s + (1.86 + 0.5i)37-s + (0.133 − 0.5i)41-s + (0.5 − 0.133i)45-s + (0.866 − 0.5i)49-s − 1.73·53-s + (−0.866 + 1.5i)61-s + (0.133 − 0.5i)65-s + (0.366 − 0.366i)73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.397333365\)
\(L(\frac12)\) \(\approx\) \(1.397333365\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.804922865738668068949974577339, −7.81930853931834974129958093556, −7.42634027557213795910589079366, −6.23946796379017680552490605722, −6.03686559884057393533404134928, −4.92615808971591766157543867925, −4.01943617850571752410834136940, −3.17851129420301611815566474307, −2.27400480838727206858161183571, −0.949962059468041605744882648822, 1.34557565107883934846665951597, 2.22631360937956780647148199734, 3.29826448814254862498395231933, 4.43229969603270613474667190166, 4.98483639816320939914246738509, 5.75989640246653950955135203822, 6.67016044597701645307563698273, 7.56454219201968016409683084225, 7.87973622927096462436235578651, 9.142359213191344357236421248720

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