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Properties

Label 2-3564-396.131-c0-0-3
Degree $2$
Conductor $3564$
Sign $0.642 + 0.766i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s + 2·19-s + (−0.499 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 0.999·28-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s + 2·19-s + (−0.499 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 0.999·28-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(1.77866\)
Root analytic conductor: \(1.33366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9834110781\)
\(L(\frac12)\) \(\approx\) \(0.9834110781\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.718903533142443597172909886796, −8.171351886443126769269018010418, −7.42626713462819022428240436774, −6.46807157490996758482872379582, −5.36842976110475696048503718374, −4.87522248381454400654359153107, −3.75018620289687586354794201735, −2.85333225347913172439151088721, −2.21692137769178557451315161072, −0.915155543048910001016215819526, 1.01483136962628947212390915908, 2.09053745033031400068304268743, 3.56029270309599393221131169895, 4.51651189331571737248799960746, 5.13928796862608842733363098349, 5.83952104640373669293700482300, 7.04277646554268681983404081592, 7.29744320363830787183967044688, 7.87625309732666033235365435320, 8.774782232999265931733329060247

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