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Properties

Label 2-3328-104.3-c0-0-4
Degree $2$
Conductor $3328$
Sign $0.869 + 0.494i$
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

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

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.869 + 0.494i$
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: \((2,\ 3328,\ (\ :0),\ 0.869 + 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254093171\)
\(L(\frac12)\) \(\approx\) \(1.254093171\)
\(L(1)\) 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

−8.740430746063027896448544934985, −8.223395457032571307483768892713, −7.30120636250685569020393299486, −6.44650523399569390783242943909, −5.74050736225294880492541245178, −4.78475466112750405639809294517, −3.90664765702914593536023884222, −3.28269642916470349026017912397, −2.59360408072397854014804456360, −0.73611963208662460745130436822, 1.41251464315648280897802188531, 2.46184413449579557827337075054, 2.93201758077765804698330347484, 4.31298621404986650738472242377, 4.95403443550666664874648975151, 6.21507828151842663547234437648, 6.60804702589260965650018733369, 7.49608531104022868534519933258, 7.910920529976933141380709374658, 8.957034722534753151835108564377

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