Properties

Label 2-8280-1.1-c1-0-16
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 − 4.77·7-s − 3.30·11-s + 3.19·13-s + 2.41·17-s + 2.24·19-s − 23-s + 25-s + 7.96·29-s − 10.3·31-s − 4.77·35-s − 11.2·37-s − 1.02·41-s + 10.0·43-s − 2.74·47-s + 15.8·49-s + 5.58·53-s − 3.30·55-s + 1.96·59-s − 8.24·61-s + 3.19·65-s − 7.71·67-s − 16.0·71-s − 7.92·73-s + 15.8·77-s + 12.5·83-s + 2.41·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.80·7-s − 0.997·11-s + 0.884·13-s + 0.585·17-s + 0.515·19-s − 0.208·23-s + 0.200·25-s + 1.47·29-s − 1.85·31-s − 0.807·35-s − 1.85·37-s − 0.160·41-s + 1.53·43-s − 0.400·47-s + 2.26·49-s + 0.767·53-s − 0.446·55-s + 0.256·59-s − 1.05·61-s + 0.395·65-s − 0.943·67-s − 1.90·71-s − 0.927·73-s + 1.80·77-s + 1.38·83-s + 0.261·85-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: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.333700313\)
\(L(\frac12)\) \(\approx\) \(1.333700313\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 + 3.30T + 11T^{2} \)
13 \( 1 - 3.19T + 13T^{2} \)
17 \( 1 - 2.41T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
29 \( 1 - 7.96T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + 1.02T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 2.74T + 47T^{2} \)
53 \( 1 - 5.58T + 53T^{2} \)
59 \( 1 - 1.96T + 59T^{2} \)
61 \( 1 + 8.24T + 61T^{2} \)
67 \( 1 + 7.71T + 67T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + 7.92T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 + 5.43T + 97T^{2} \)
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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

−7.62523578504340344127715946882, −7.12426793746826750400411119482, −6.32620635415832967358459785262, −5.79985145912396292439848601502, −5.26026551281995224423491400092, −4.13433753112427157699608144314, −3.24640860908209816468680139409, −2.94778673171720573651073140046, −1.79015294388898879380917452127, −0.55262943091967309427702018438, 0.55262943091967309427702018438, 1.79015294388898879380917452127, 2.94778673171720573651073140046, 3.24640860908209816468680139409, 4.13433753112427157699608144314, 5.26026551281995224423491400092, 5.79985145912396292439848601502, 6.32620635415832967358459785262, 7.12426793746826750400411119482, 7.62523578504340344127715946882

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