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Properties

Label 2-3328-13.2-c0-0-0
Degree $2$
Conductor $3328$
Sign $0.477 + 0.878i$
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  + (−1.36 + 1.36i)5-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)13-s + (−0.866 − 0.5i)17-s − 2.73i·25-s + (−0.5 − 0.866i)29-s + (0.133 − 0.5i)37-s + (1.86 + 0.5i)41-s + (0.499 + 1.86i)45-s + (−0.866 + 0.5i)49-s + 1.73·53-s + (0.866 − 1.5i)61-s + (1.86 + 0.499i)65-s + (−1.36 − 1.36i)73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)5-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)13-s + (−0.866 − 0.5i)17-s − 2.73i·25-s + (−0.5 − 0.866i)29-s + (0.133 − 0.5i)37-s + (1.86 + 0.5i)41-s + (0.499 + 1.86i)45-s + (−0.866 + 0.5i)49-s + 1.73·53-s + (0.866 − 1.5i)61-s + (1.86 + 0.499i)65-s + (−1.36 − 1.36i)73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.477 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6988561616\)
\(L(\frac12)\) \(\approx\) \(0.6988561616\)
\(L(1)\) 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 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
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 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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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

−8.536693929678438528421834699501, −7.69007884446542110387596390438, −7.30025502514073888703887930609, −6.60865347225230544784029550473, −5.87056096117536623227498088898, −4.54700064586347361246739811171, −3.95307248678726845271874443675, −3.14509735990232400271127226609, −2.41279426974263457945920326051, −0.45957740783345809860842734674, 1.22930708743613414738356180838, 2.31211019636281966164021052760, 3.78798766434395178226370928750, 4.35290587035079328923646873532, 4.87504889333634073026327322012, 5.72968425538747699846814100811, 7.10340501780488681799603617844, 7.37418147350973145143368214829, 8.339796126360831945034308972325, 8.733561479069540471978973292331

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