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Properties

Label 2-3328-104.35-c0-0-0
Degree $2$
Conductor $3328$
Sign $0.00641 - 0.999i$
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  + i·5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)29-s + (−0.866 − 0.5i)37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)45-s + (−0.5 + 0.866i)49-s + i·53-s + (−0.866 + 0.5i)61-s + (0.5 − 0.866i)65-s + 73-s + (−0.499 + 0.866i)81-s + (−0.866 + 0.5i)85-s + ⋯
L(s)  = 1  + i·5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)29-s + (−0.866 − 0.5i)37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)45-s + (−0.5 + 0.866i)49-s + i·53-s + (−0.866 + 0.5i)61-s + (0.5 − 0.866i)65-s + 73-s + (−0.499 + 0.866i)81-s + (−0.866 + 0.5i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.00641 - 0.999i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.00641 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178890267\)
\(L(\frac12)\) \(\approx\) \(1.178890267\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - iT - T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.965653733124425956221117388846, −8.045085640810775741929021238355, −7.51944552108666692626020858816, −6.85220096274287659064030851142, −6.06569267911931925746114011186, −5.15696655505296593124735188219, −4.42907957730373166561845414079, −3.31368379652983501730193352783, −2.63373456184099927720360864539, −1.56599502301287461660661888649, 0.72639726108826262235582461095, 1.87213207380416132541205380326, 3.08745135995926736441269617112, 4.04440052172073897313455942575, 4.86960637037317231447509585896, 5.35412220574970014388231748305, 6.56905632179135274999819246610, 7.00986898183359277350140644475, 7.984978516981323511946726742635, 8.657715583656399842839678897590

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