Properties

Label 2-8280-1.1-c1-0-19
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 + 2.89·7-s − 3.43·11-s − 4.72·13-s − 2.61·17-s + 2.28·19-s + 23-s + 25-s − 2.53·29-s + 6.19·31-s − 2.89·35-s + 4.89·37-s − 10.4·41-s − 9.01·43-s + 2.28·47-s + 1.40·49-s − 0.682·53-s + 3.43·55-s + 14.2·59-s + 4.98·61-s + 4.72·65-s + 1.74·67-s + 6.32·71-s − 1.51·73-s − 9.94·77-s − 3.86·79-s + 4.61·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.09·7-s − 1.03·11-s − 1.31·13-s − 0.634·17-s + 0.523·19-s + 0.208·23-s + 0.200·25-s − 0.470·29-s + 1.11·31-s − 0.489·35-s + 0.805·37-s − 1.63·41-s − 1.37·43-s + 0.332·47-s + 0.200·49-s − 0.0936·53-s + 0.462·55-s + 1.85·59-s + 0.638·61-s + 0.586·65-s + 0.213·67-s + 0.751·71-s − 0.177·73-s − 1.13·77-s − 0.434·79-s + 0.506·83-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

\(L(1)\) \(\approx\) \(1.505523889\)
\(L(\frac12)\) \(\approx\) \(1.505523889\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.89T + 7T^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 + 4.72T + 13T^{2} \)
17 \( 1 + 2.61T + 17T^{2} \)
19 \( 1 - 2.28T + 19T^{2} \)
29 \( 1 + 2.53T + 29T^{2} \)
31 \( 1 - 6.19T + 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 9.01T + 43T^{2} \)
47 \( 1 - 2.28T + 47T^{2} \)
53 \( 1 + 0.682T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 - 1.74T + 67T^{2} \)
71 \( 1 - 6.32T + 71T^{2} \)
73 \( 1 + 1.51T + 73T^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 - 4.61T + 83T^{2} \)
89 \( 1 - 6.31T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−7.900665255305339178569143937760, −7.21941495570227934432946211200, −6.61014527591542051501988910941, −5.45250814979711014299741836143, −4.96183163639127106591683449240, −4.52336351759448119970158099903, −3.46263438954715637089913477911, −2.56848309304838944365447128978, −1.89097787696178421380325412060, −0.58641742493072488737702014572, 0.58641742493072488737702014572, 1.89097787696178421380325412060, 2.56848309304838944365447128978, 3.46263438954715637089913477911, 4.52336351759448119970158099903, 4.96183163639127106591683449240, 5.45250814979711014299741836143, 6.61014527591542051501988910941, 7.21941495570227934432946211200, 7.900665255305339178569143937760

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