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Properties

Label 2-3328-104.5-c0-0-1
Degree $2$
Conductor $3328$
Sign $0.881 + 0.471i$
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  + (−1 − i)5-s + 9-s + 13-s + 2i·17-s + i·25-s + (1 − i)37-s + (1 − i)41-s + (−1 − i)45-s − i·49-s + 2i·53-s − 2i·61-s + (−1 − i)65-s + (−1 − i)73-s + 81-s + (2 − 2i)85-s + ⋯
L(s)  = 1  + (−1 − i)5-s + 9-s + 13-s + 2i·17-s + i·25-s + (1 − i)37-s + (1 − i)41-s + (−1 − i)45-s − i·49-s + 2i·53-s − 2i·61-s + (−1 − i)65-s + (−1 − i)73-s + 81-s + (2 − 2i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191973354\)
\(L(\frac12)\) \(\approx\) \(1.191973354\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 - T^{2} \)
5 \( 1 + (1 + i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−8.749954762016054540404361947249, −7.942486595257459185274748125390, −7.56009940926107554137546572006, −6.43335466222967252598831935893, −5.78910303938617192950743591993, −4.69437785965434527808122641406, −4.03404855273705876364676430186, −3.61631415258132266680007541226, −1.93998072963048092232202070219, −0.980288720884505596464552726733, 1.06768287165927724189489965434, 2.59972342937837816952346250843, 3.31816991109353279586673098472, 4.17889105252932084025410923087, 4.84323051948821743257578545388, 6.04648315809939378658111945797, 6.81221192525764554607565688903, 7.36891960002028428987148474131, 7.893629960472062535142842068210, 8.836392189814190760479526722630

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