Properties

Label 2-8280-1.1-c1-0-23
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 + 0.121·7-s − 2.87·11-s + 5.22·13-s + 2.22·17-s + 1.22·19-s − 23-s + 25-s − 9.34·29-s − 2.12·31-s − 0.121·35-s + 5.59·37-s − 8.22·41-s + 8·43-s + 10.4·47-s − 6.98·49-s + 3.59·53-s + 2.87·55-s + 0.650·59-s − 7.33·61-s − 5.22·65-s + 5.59·67-s + 13.9·71-s + 12.9·73-s − 0.349·77-s − 3.51·79-s + 11.1·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.0459·7-s − 0.867·11-s + 1.45·13-s + 0.540·17-s + 0.281·19-s − 0.208·23-s + 0.200·25-s − 1.73·29-s − 0.381·31-s − 0.0205·35-s + 0.919·37-s − 1.28·41-s + 1.21·43-s + 1.52·47-s − 0.997·49-s + 0.493·53-s + 0.388·55-s + 0.0846·59-s − 0.939·61-s − 0.648·65-s + 0.683·67-s + 1.65·71-s + 1.51·73-s − 0.0398·77-s − 0.395·79-s + 1.21·83-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: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.730249888\)
\(L(\frac12)\) \(\approx\) \(1.730249888\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.121T + 7T^{2} \)
11 \( 1 + 2.87T + 11T^{2} \)
13 \( 1 - 5.22T + 13T^{2} \)
17 \( 1 - 2.22T + 17T^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
29 \( 1 + 9.34T + 29T^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 - 5.59T + 37T^{2} \)
41 \( 1 + 8.22T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 3.59T + 53T^{2} \)
59 \( 1 - 0.650T + 59T^{2} \)
61 \( 1 + 7.33T + 61T^{2} \)
67 \( 1 - 5.59T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 3.51T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 0.486T + 89T^{2} \)
97 \( 1 - 0.635T + 97T^{2} \)
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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

−7.933941265653865905993035050915, −7.22639220921320099555227471987, −6.41010998119142004645101667310, −5.63013345753158074281748672730, −5.19332366885526801019576690191, −4.04440243182723350194702502142, −3.64142121406332749149034107718, −2.73133099074177703330270970874, −1.71872603270270747772551083490, −0.65559402291503074044996035287, 0.65559402291503074044996035287, 1.71872603270270747772551083490, 2.73133099074177703330270970874, 3.64142121406332749149034107718, 4.04440243182723350194702502142, 5.19332366885526801019576690191, 5.63013345753158074281748672730, 6.41010998119142004645101667310, 7.22639220921320099555227471987, 7.933941265653865905993035050915

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