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Properties

Label 2-2816-44.27-c0-0-0
Degree $2$
Conductor $2816$
Sign $0.624 - 0.781i$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
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  + (0.363 − 0.5i)3-s + (0.190 + 0.587i)9-s + (−0.587 + 0.809i)11-s + (−0.5 + 1.53i)17-s + (−0.951 + 1.30i)19-s + (0.809 + 0.587i)25-s + (0.951 + 0.309i)27-s + (0.190 + 0.587i)33-s + (0.5 + 0.363i)41-s − 0.618i·43-s + (0.309 − 0.951i)49-s + (0.587 + 0.809i)51-s + (0.309 + 0.951i)57-s + (−0.951 − 1.30i)59-s − 1.61i·67-s + ⋯
L(s)  = 1  + (0.363 − 0.5i)3-s + (0.190 + 0.587i)9-s + (−0.587 + 0.809i)11-s + (−0.5 + 1.53i)17-s + (−0.951 + 1.30i)19-s + (0.809 + 0.587i)25-s + (0.951 + 0.309i)27-s + (0.190 + 0.587i)33-s + (0.5 + 0.363i)41-s − 0.618i·43-s + (0.309 − 0.951i)49-s + (0.587 + 0.809i)51-s + (0.309 + 0.951i)57-s + (−0.951 − 1.30i)59-s − 1.61i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $0.624 - 0.781i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2816} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2816,\ (\ :0),\ 0.624 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.221304916\)
\(L(\frac12)\) \(\approx\) \(1.221304916\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (0.587 - 0.809i)T \)
good3 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618iT - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + 1.61iT - T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.899640880581766225503152767298, −8.162381157819591615220895886209, −7.76296740992445856374537258844, −6.85753557939880150060666326133, −6.17409101699563530273242026171, −5.17600499835200992887782535807, −4.37902936955986319596061627843, −3.45745653618966451210814018508, −2.19732347567155803295540137445, −1.69145008582169972274546652634, 0.73490764131039749142186403868, 2.58737391304007558095990918835, 3.01323645132359898335029052113, 4.25912453617585859467222676165, 4.76233334584279314096324136778, 5.79510646081333663493903251058, 6.67744545188377474959641349177, 7.28270058744641769359449587531, 8.347985552579332678930253256515, 8.957138581310777209510159748038

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