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Properties

Label 2-3328-104.85-c0-0-1
Degree $2$
Conductor $3328$
Sign $0.283 + 0.958i$
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  + (0.366 − 0.366i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.866 − 0.5i)17-s + 0.732i·25-s + (−1.5 − 0.866i)29-s + (−1.86 + 0.5i)37-s + (1.86 − 0.5i)41-s + (−0.5 − 0.133i)45-s + (−0.866 − 0.5i)49-s − i·53-s + (0.866 − 0.5i)61-s + (−0.133 − 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (0.366 − 0.366i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.866 − 0.5i)17-s + 0.732i·25-s + (−1.5 − 0.866i)29-s + (−1.86 + 0.5i)37-s + (1.86 − 0.5i)41-s + (−0.5 − 0.133i)45-s + (−0.866 − 0.5i)49-s − i·53-s + (0.866 − 0.5i)61-s + (−0.133 − 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.283 + 0.958i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.283 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240797971\)
\(L(\frac12)\) \(\approx\) \(1.240797971\)
\(L(1)\) 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 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
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 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + iT - T^{2} \)
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 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
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

−8.711398981065079343866128561323, −7.960162431337252185807076849113, −7.22285804697962889266174896650, −6.26485077718987502282258671847, −5.60157827307155737785397712986, −5.07650242769767254976628759390, −3.72048342350847623366817525846, −3.27027318632784256529949984770, −2.00359710738641751222314370834, −0.76009050882679437910665504439, 1.55275188061677362922692401739, 2.41165000993393497668636566278, 3.43123527672051933819983800130, 4.28604815814138452805054339504, 5.33594934651065302878081337204, 5.87089796064282651685902778272, 6.71650543991791644208186699983, 7.50150163613653361597015490515, 8.204138262500032275404458138142, 8.967290125692116922375920852404

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