Properties

Label 2-8280-1.1-c1-0-46
Degree $2$
Conductor $8280$
Sign $1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 0.476·7-s + 1.67·11-s + 2.67·13-s − 1.67·17-s + 7.92·19-s − 23-s + 25-s + 2.20·29-s + 6.77·31-s + 0.476·35-s + 4·37-s − 5.97·41-s − 0.402·43-s − 1.79·47-s − 6.77·49-s + 10.8·53-s + 1.67·55-s − 9.45·59-s + 6.32·61-s + 2.67·65-s + 14.7·67-s − 9.97·71-s − 6.10·73-s + 0.798·77-s − 4.49·83-s − 1.67·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.179·7-s + 0.505·11-s + 0.742·13-s − 0.406·17-s + 1.81·19-s − 0.208·23-s + 0.200·25-s + 0.408·29-s + 1.21·31-s + 0.0804·35-s + 0.657·37-s − 0.933·41-s − 0.0614·43-s − 0.262·47-s − 0.967·49-s + 1.48·53-s + 0.226·55-s − 1.23·59-s + 0.809·61-s + 0.332·65-s + 1.79·67-s − 1.18·71-s − 0.714·73-s + 0.0910·77-s − 0.493·83-s − 0.181·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.776476160\)
\(L(\frac12)\) \(\approx\) \(2.776476160\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 - 0.476T + 7T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 - 2.67T + 13T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 - 7.92T + 19T^{2} \)
29 \( 1 - 2.20T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 5.97T + 41T^{2} \)
43 \( 1 + 0.402T + 43T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 9.45T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 9.97T + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.49T + 83T^{2} \)
89 \( 1 - 3.59T + 89T^{2} \)
97 \( 1 + 6.08T + 97T^{2} \)
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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

−7.86537482083663431555632924545, −7.01915852237603938630313080815, −6.44471021288025615794610154130, −5.75454344262708071395036682699, −5.06165794925888952068682252568, −4.30340494636804541825034726831, −3.43787744658918511573437266665, −2.72035460798698210796465671434, −1.63590201577779577115414381602, −0.884356630413398266102078288759, 0.884356630413398266102078288759, 1.63590201577779577115414381602, 2.72035460798698210796465671434, 3.43787744658918511573437266665, 4.30340494636804541825034726831, 5.06165794925888952068682252568, 5.75454344262708071395036682699, 6.44471021288025615794610154130, 7.01915852237603938630313080815, 7.86537482083663431555632924545

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