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Properties

Label 2-3564-297.175-c0-0-0
Degree $2$
Conductor $3564$
Sign $-0.396 - 0.918i$
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  + (−1.17 − 0.984i)5-s + (−0.766 + 0.642i)11-s + (−0.939 + 0.342i)23-s + (0.233 + 1.32i)25-s + (−1.43 + 0.524i)31-s + (−0.173 + 0.300i)37-s + (1.43 + 0.524i)47-s + (0.766 + 0.642i)49-s − 0.347·53-s + 1.53·55-s + (−1.17 − 0.984i)59-s + (−0.326 + 1.85i)67-s + (−0.939 + 1.62i)71-s + (1 + 1.73i)89-s + (−1.43 + 1.20i)97-s + ⋯
L(s)  = 1  + (−1.17 − 0.984i)5-s + (−0.766 + 0.642i)11-s + (−0.939 + 0.342i)23-s + (0.233 + 1.32i)25-s + (−1.43 + 0.524i)31-s + (−0.173 + 0.300i)37-s + (1.43 + 0.524i)47-s + (0.766 + 0.642i)49-s − 0.347·53-s + 1.53·55-s + (−1.17 − 0.984i)59-s + (−0.326 + 1.85i)67-s + (−0.939 + 1.62i)71-s + (1 + 1.73i)89-s + (−1.43 + 1.20i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.396 - 0.918i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.396 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3270542047\)
\(L(\frac12)\) \(\approx\) \(0.3270542047\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
good5 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 + 0.347T + T^{2} \)
59 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
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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

−8.882644521693448887401333974345, −8.151157500615827314333484174355, −7.61260293883570264651328321622, −7.02147622229617431919847296759, −5.78810664244011487399756327024, −5.13100649650043313739477408840, −4.32066615798217433352711760728, −3.76010241867719506057956818964, −2.58383275469324306935633369505, −1.35412497196046197178837449931, 0.19286607932210421663912062784, 2.10739341477562738469159928605, 3.08977141700344352256327177970, 3.71536119488959436072418531693, 4.52251926633362053751363839881, 5.63091051160274758776985144515, 6.26652403584444331196214931895, 7.36064868667028987709512552479, 7.54778730425949233323817372381, 8.389470884448596950214118224728

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