Properties

Label 2-8280-1.1-c1-0-103
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 3.44·7-s − 2.44·11-s − 4.44·13-s − 0.550·17-s + 2.44·19-s − 23-s + 25-s − 7.89·29-s − 7.89·31-s + 3.44·35-s + 8.34·37-s − 1.89·41-s − 0.898·43-s − 1.55·47-s + 4.89·49-s + 7.44·53-s − 2.44·55-s − 59-s + 4.44·61-s − 4.44·65-s − 9.44·67-s − 5.89·71-s − 3.55·73-s − 8.44·77-s − 15.4·83-s − 0.550·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.30·7-s − 0.738·11-s − 1.23·13-s − 0.133·17-s + 0.561·19-s − 0.208·23-s + 0.200·25-s − 1.46·29-s − 1.41·31-s + 0.583·35-s + 1.37·37-s − 0.296·41-s − 0.137·43-s − 0.226·47-s + 0.699·49-s + 1.02·53-s − 0.330·55-s − 0.130·59-s + 0.569·61-s − 0.551·65-s − 1.15·67-s − 0.700·71-s − 0.415·73-s − 0.962·77-s − 1.69·83-s − 0.0597·85-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: \(1\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + 0.550T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
29 \( 1 + 7.89T + 29T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 - 8.34T + 37T^{2} \)
41 \( 1 + 1.89T + 41T^{2} \)
43 \( 1 + 0.898T + 43T^{2} \)
47 \( 1 + 1.55T + 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 + T + 59T^{2} \)
61 \( 1 - 4.44T + 61T^{2} \)
67 \( 1 + 9.44T + 67T^{2} \)
71 \( 1 + 5.89T + 71T^{2} \)
73 \( 1 + 3.55T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 8.89T + 89T^{2} \)
97 \( 1 + 1.10T + 97T^{2} \)
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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

−7.45533438511854099882657417762, −7.05074059766755066945718435990, −5.76237648170867533742697236550, −5.44018893749979220373850561463, −4.75064185031391896362260835966, −4.04792494046729827266251847162, −2.89413608040190366073225310833, −2.16394552726948847362326492208, −1.44476742938315639647106323083, 0, 1.44476742938315639647106323083, 2.16394552726948847362326492208, 2.89413608040190366073225310833, 4.04792494046729827266251847162, 4.75064185031391896362260835966, 5.44018893749979220373850561463, 5.76237648170867533742697236550, 7.05074059766755066945718435990, 7.45533438511854099882657417762

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