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Properties

Label 2-3328-104.45-c0-0-0
Degree $2$
Conductor $3328$
Sign $0.295 - 0.955i$
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 + (1.5 + 0.866i)29-s + (0.133 + 0.5i)37-s + (0.133 + 0.5i)41-s + (0.499 − 1.86i)45-s + (0.866 + 0.5i)49-s − i·53-s + (0.866 − 0.5i)61-s + (−1.86 + 0.499i)65-s + (−0.366 − 0.366i)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 + (1.5 + 0.866i)29-s + (0.133 + 0.5i)37-s + (0.133 + 0.5i)41-s + (0.499 − 1.86i)45-s + (0.866 + 0.5i)49-s − i·53-s + (0.866 − 0.5i)61-s + (−1.86 + 0.499i)65-s + (−0.366 − 0.366i)73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.295 - 0.955i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.295 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.455932451\)
\(L(\frac12)\) \(\approx\) \(1.455932451\)
\(L(1)\) 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 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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

−9.094625951448024448834513258631, −8.367961379575979254826563609232, −7.06976112393957234715857435112, −6.68297666436327708025664872979, −6.16973630695720549709955745130, −5.35701155000773536628563734028, −4.28303909530734965116018440426, −3.17581378630306381035729390321, −2.55769374648017486053446027619, −1.62234241986152846072797458785, 0.867005665124100306582583730909, 2.18185987473414201355319546283, 2.65883080894049094210515550327, 4.32070130265477081095739799137, 4.95276314727428422449442762680, 5.55500193687132870116964110503, 6.14298998455370009298131736416, 7.20690629043680283261736461526, 8.180528002666949177244238925556, 8.650582896973925667190453143930

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