Properties

Label 2-8280-1.1-c1-0-104
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 + 3.13·7-s − 2.65·11-s + 0.651·13-s − 4.48·17-s − 0.651·19-s − 23-s + 25-s − 2.48·29-s − 3.13·31-s + 3.13·35-s − 6.10·37-s − 0.820·41-s + 0.696·43-s + 11.8·47-s + 2.83·49-s − 9.45·53-s − 2.65·55-s − 12.7·59-s + 2.65·61-s + 0.651·65-s − 6.10·67-s − 8.48·71-s + 6.31·73-s − 8.31·77-s − 8·79-s − 5.51·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.18·7-s − 0.799·11-s + 0.180·13-s − 1.08·17-s − 0.149·19-s − 0.208·23-s + 0.200·25-s − 0.461·29-s − 0.563·31-s + 0.530·35-s − 1.00·37-s − 0.128·41-s + 0.106·43-s + 1.73·47-s + 0.404·49-s − 1.29·53-s − 0.357·55-s − 1.66·59-s + 0.339·61-s + 0.0808·65-s − 0.745·67-s − 1.00·71-s + 0.739·73-s − 0.947·77-s − 0.900·79-s − 0.605·83-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: \(1\)
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 \( 1 - 3.13T + 7T^{2} \)
11 \( 1 + 2.65T + 11T^{2} \)
13 \( 1 - 0.651T + 13T^{2} \)
17 \( 1 + 4.48T + 17T^{2} \)
19 \( 1 + 0.651T + 19T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 + 3.13T + 31T^{2} \)
37 \( 1 + 6.10T + 37T^{2} \)
41 \( 1 + 0.820T + 41T^{2} \)
43 \( 1 - 0.696T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + 9.45T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 - 2.65T + 61T^{2} \)
67 \( 1 + 6.10T + 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 - 6.31T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 5.51T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 5.93T + 97T^{2} \)
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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

−7.55777162129732734266336237546, −6.82181435257892710733971891835, −5.97844226930335292282344444067, −5.34702088235022636661906045224, −4.71216434801486076305088088163, −4.04840877051860323186689387777, −2.94753007686425542547669341764, −2.10319994057678539398627559737, −1.47123120874092272055658155091, 0, 1.47123120874092272055658155091, 2.10319994057678539398627559737, 2.94753007686425542547669341764, 4.04840877051860323186689387777, 4.71216434801486076305088088163, 5.34702088235022636661906045224, 5.97844226930335292282344444067, 6.82181435257892710733971891835, 7.55777162129732734266336237546

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