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Properties

Label 2-3564-396.263-c0-0-6
Degree $2$
Conductor $3564$
Sign $-0.173 + 0.984i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.22 − 0.707i)5-s + (0.707 − 1.22i)7-s − 0.999i·8-s − 1.41·10-s + (−0.866 − 0.5i)11-s + (−1.22 + 0.707i)14-s + (−0.5 + 0.866i)16-s + 1.41·19-s + (1.22 + 0.707i)20-s + (0.499 + 0.866i)22-s + (0.499 − 0.866i)25-s + 1.41·28-s + (0.866 − 0.499i)32-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.22 − 0.707i)5-s + (0.707 − 1.22i)7-s − 0.999i·8-s − 1.41·10-s + (−0.866 − 0.5i)11-s + (−1.22 + 0.707i)14-s + (−0.5 + 0.866i)16-s + 1.41·19-s + (1.22 + 0.707i)20-s + (0.499 + 0.866i)22-s + (0.499 − 0.866i)25-s + 1.41·28-s + (0.866 − 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (2375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132711040\)
\(L(\frac12)\) \(\approx\) \(1.132711040\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
11 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.545087676581473854369385441683, −7.938703363852583270771135084640, −7.33131949408067008854994526809, −6.43462060033657364969244782113, −5.43325187322222061114812605947, −4.79227622323708556297244948249, −3.69939775286984402145275890696, −2.72613272003005340675635453383, −1.64551758909704475439122040972, −0.934391101039654514184864222268, 1.58166444190673571556988340718, 2.30074306478622129461559896916, 2.97064614066205600132159583017, 4.85054833153183912883911159281, 5.50226595132038005391093568159, 5.87462419332294166120834790500, 6.81295667532441392204653566879, 7.51523984621900607300709156358, 8.198471097129770145311735599785, 9.070194314638093176493234700985

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