Properties

Label 2-8280-1.1-c1-0-3
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 − 2.37·7-s − 4·11-s − 6.74·13-s + 0.372·17-s + 4·19-s − 23-s + 25-s − 0.372·29-s − 2.37·31-s + 2.37·35-s + 0.372·37-s − 9.11·41-s − 4·43-s − 4.74·47-s − 1.37·49-s + 4.37·53-s + 4·55-s − 14.3·59-s − 2·61-s + 6.74·65-s + 9.62·67-s + 2.37·71-s − 1.25·73-s + 9.48·77-s + 9.48·79-s − 15.8·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.896·7-s − 1.20·11-s − 1.87·13-s + 0.0902·17-s + 0.917·19-s − 0.208·23-s + 0.200·25-s − 0.0691·29-s − 0.426·31-s + 0.400·35-s + 0.0612·37-s − 1.42·41-s − 0.609·43-s − 0.692·47-s − 0.196·49-s + 0.600·53-s + 0.539·55-s − 1.87·59-s − 0.256·61-s + 0.836·65-s + 1.17·67-s + 0.281·71-s − 0.146·73-s + 1.08·77-s + 1.06·79-s − 1.74·83-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

\(L(1)\) \(\approx\) \(0.4769175729\)
\(L(\frac12)\) \(\approx\) \(0.4769175729\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 6.74T + 13T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 + 0.372T + 29T^{2} \)
31 \( 1 + 2.37T + 31T^{2} \)
37 \( 1 - 0.372T + 37T^{2} \)
41 \( 1 + 9.11T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 4.74T + 47T^{2} \)
53 \( 1 - 4.37T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 9.62T + 67T^{2} \)
71 \( 1 - 2.37T + 71T^{2} \)
73 \( 1 + 1.25T + 73T^{2} \)
79 \( 1 - 9.48T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 6.74T + 89T^{2} \)
97 \( 1 + 7.48T + 97T^{2} \)
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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

−7.78528995257728834276143300479, −7.13185210646970560630039252648, −6.63659154652459136049439924999, −5.52096834867068834183701973024, −5.12167129017756136471920067878, −4.34407450425872949378822075948, −3.23733522493577617190370047874, −2.88501937440715585415500182619, −1.88060025084583719953162132411, −0.31743108799320733658050039418, 0.31743108799320733658050039418, 1.88060025084583719953162132411, 2.88501937440715585415500182619, 3.23733522493577617190370047874, 4.34407450425872949378822075948, 5.12167129017756136471920067878, 5.52096834867068834183701973024, 6.63659154652459136049439924999, 7.13185210646970560630039252648, 7.78528995257728834276143300479

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