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Properties

Label 2-3564-99.14-c0-0-0
Degree $2$
Conductor $3564$
Sign $0.264 - 0.964i$
Analytic cond. $1.77866$
Root an. cond. $1.33366$
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  + (0.251 + 0.564i)5-s + (−0.978 + 0.207i)7-s + (0.406 − 0.913i)11-s + (0.0646 + 0.614i)13-s + (−0.951 + 1.30i)17-s + (−0.309 − 0.951i)19-s + (1.40 + 0.809i)23-s + (0.413 − 0.459i)25-s + (0.207 + 0.978i)29-s + (0.104 + 0.994i)31-s + (−0.363 − 0.5i)35-s + (−0.207 + 0.978i)41-s + (0.459 + 0.413i)47-s + (0.587 + 0.809i)53-s + 0.618·55-s + ⋯
L(s)  = 1  + (0.251 + 0.564i)5-s + (−0.978 + 0.207i)7-s + (0.406 − 0.913i)11-s + (0.0646 + 0.614i)13-s + (−0.951 + 1.30i)17-s + (−0.309 − 0.951i)19-s + (1.40 + 0.809i)23-s + (0.413 − 0.459i)25-s + (0.207 + 0.978i)29-s + (0.104 + 0.994i)31-s + (−0.363 − 0.5i)35-s + (−0.207 + 0.978i)41-s + (0.459 + 0.413i)47-s + (0.587 + 0.809i)53-s + 0.618·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $0.264 - 0.964i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ 0.264 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.079491883\)
\(L(\frac12)\) \(\approx\) \(1.079491883\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-0.406 + 0.913i)T \)
good5 \( 1 + (-0.251 - 0.564i)T + (-0.669 + 0.743i)T^{2} \)
7 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
13 \( 1 + (-0.0646 - 0.614i)T + (-0.978 + 0.207i)T^{2} \)
17 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-1.40 - 0.809i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \)
31 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.207 - 0.978i)T + (-0.913 - 0.406i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.459 - 0.413i)T + (0.104 + 0.994i)T^{2} \)
53 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2} \)
61 \( 1 + (-0.0646 + 0.614i)T + (-0.978 - 0.207i)T^{2} \)
67 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.669 + 0.743i)T^{2} \)
83 \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \)
89 \( 1 + 0.618iT - T^{2} \)
97 \( 1 + (0.669 + 0.743i)T^{2} \)
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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

−8.970747466405574114088923204811, −8.381571561196318510831462259260, −7.11869802113047295776412104179, −6.56747234539185956232909145650, −6.22169674659392250781787609694, −5.16698549622622209477388552632, −4.19545092579718652225760235371, −3.24967600516250510488907008431, −2.69352373173770047887723530321, −1.35923468168576949616787194889, 0.66124821136193799834364908935, 2.08681104161121410729319029107, 2.98980604725806464368739878375, 4.01670347087667567536421161103, 4.76804176643157058401329926583, 5.53599879572407718388235642612, 6.50364910474205885674424402791, 6.98492702469983737479002279569, 7.79518650826277844498087308895, 8.808889548882894248699709863082

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