Properties

Label 2-2816-44.27-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.142701155\)
\(L(\frac12)\) \(\approx\) \(1.142701155\)
\(L(1)\) 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

−8.984389005992329607270361613078, −8.523665750692238107623802558185, −7.54344313949232591825203523556, −6.77744573682330167348706633703, −5.92856256932458630384681509770, −5.22027280169041745715354945723, −4.39636456110711061458654572053, −3.62094783270972796528766668595, −2.55310026542305238196571374968, −1.24054915449849169087308295044, 0.916236904739694247434245719289, 2.03587572818094307007747942992, 3.21828601996219833552112537357, 4.15897390591405130318172172743, 5.03146318500948112379403063600, 5.89587142708707045667907735243, 6.74919734986664379145336472511, 7.18158786852728436217618135245, 7.953791168691187555661484927658, 9.036463288916524052422143308024

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