Properties

Degree $2$
Conductor $816$
Sign $0.970 - 0.242i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + 4i·7-s − 9-s − 4i·11-s + 2·13-s + (1 + 4i)17-s + 4·19-s + 4·21-s + 4i·23-s + 5·25-s + i·27-s + 4i·31-s − 4·33-s + 8i·37-s − 2i·39-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.51i·7-s − 0.333·9-s − 1.20i·11-s + 0.554·13-s + (0.242 + 0.970i)17-s + 0.917·19-s + 0.872·21-s + 0.834i·23-s + 25-s + 0.192i·27-s + 0.718i·31-s − 0.696·33-s + 1.31i·37-s − 0.320i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.970 - 0.242i$
Motivic weight: \(1\)
Character: $\chi_{816} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56432 + 0.192577i\)
\(L(\frac12)\) \(\approx\) \(1.56432 + 0.192577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
17 \( 1 + (-1 - 4i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−10.36729441438693682621504955593, −9.062012836245030306475549787344, −8.677379097525106227934651201590, −7.86965371605991490770153890036, −6.68472521839852822760758759176, −5.79428079475028492709609299346, −5.33493735024098629651162152676, −3.58132618596494580605195105364, −2.68526237322172352158582426614, −1.29882060825980296171502281625, 0.946357922086703167434109693855, 2.74483378635255103417511074674, 4.02089302581035142580518405045, 4.56656649062571355611528055195, 5.68580132492195408925575053402, 7.09585719645274874954349186482, 7.34587126078453464042934746312, 8.602474759245174437297129837887, 9.602328290184690979558291126884, 10.15236340175770776494369711976

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