Properties

Label 2-8280-69.68-c1-0-63
Degree $2$
Conductor $8280$
Sign $0.997 - 0.0646i$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 1.36i·7-s + 2.32·11-s + 6.17·13-s + 6.91·17-s − 0.609i·19-s + (4.08 + 2.50i)23-s + 25-s − 0.841i·29-s + 0.0981·31-s − 1.36i·35-s − 6.85i·37-s − 6.58i·41-s − 0.293i·43-s + 3.82i·47-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.514i·7-s + 0.702·11-s + 1.71·13-s + 1.67·17-s − 0.139i·19-s + (0.852 + 0.523i)23-s + 0.200·25-s − 0.156i·29-s + 0.0176·31-s − 0.230i·35-s − 1.12i·37-s − 1.02i·41-s − 0.0446i·43-s + 0.557i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.997 - 0.0646i$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8280} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 0.997 - 0.0646i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.517443701\)
\(L(\frac12)\) \(\approx\) \(2.517443701\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.08 - 2.50i)T \)
good7 \( 1 - 1.36iT - 7T^{2} \)
11 \( 1 - 2.32T + 11T^{2} \)
13 \( 1 - 6.17T + 13T^{2} \)
17 \( 1 - 6.91T + 17T^{2} \)
19 \( 1 + 0.609iT - 19T^{2} \)
29 \( 1 + 0.841iT - 29T^{2} \)
31 \( 1 - 0.0981T + 31T^{2} \)
37 \( 1 + 6.85iT - 37T^{2} \)
41 \( 1 + 6.58iT - 41T^{2} \)
43 \( 1 + 0.293iT - 43T^{2} \)
47 \( 1 - 3.82iT - 47T^{2} \)
53 \( 1 - 1.36T + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 + 2.54iT - 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 + 4.87iT - 71T^{2} \)
73 \( 1 - 3.41T + 73T^{2} \)
79 \( 1 + 2.39iT - 79T^{2} \)
83 \( 1 - 2.10T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 7.80iT - 97T^{2} \)
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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

−7.75206500948964582000565968147, −7.24793767077458799013214576012, −6.32292873013087836039888948870, −5.77460425228865525179849124815, −5.16837963698793847402340678140, −4.03010119291472939314796648969, −3.61068747214337867205938940715, −2.83239218578612341046043972214, −1.58563840841336419320746446099, −0.841811740560332858064412689915, 0.937096635401324111936616228942, 1.38246873533019472556835211106, 2.89349779180342596719434972907, 3.59446450564592872853454683618, 4.05746802743592887421946027378, 5.00017839114575831499840533952, 5.77834297153667063318596522502, 6.51067028443928906027973075360, 7.03872469345195955535909069014, 7.909705012178947117228393437423

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