Properties

Label 2-8280-1.1-c1-0-27
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 + 1.53·7-s + 0.860·11-s + 0.139·13-s − 5.50·17-s + 5.25·19-s + 23-s + 25-s − 9.76·29-s + 6.78·31-s − 1.53·35-s − 12.0·37-s + 9.98·41-s + 11.4·43-s + 2.32·47-s − 4.63·49-s − 0.149·53-s − 0.860·55-s + 11.0·59-s + 4.43·61-s − 0.139·65-s − 10.4·67-s − 7.31·71-s + 7.11·73-s + 1.32·77-s + 6.79·79-s − 10.6·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.581·7-s + 0.259·11-s + 0.0386·13-s − 1.33·17-s + 1.20·19-s + 0.208·23-s + 0.200·25-s − 1.81·29-s + 1.21·31-s − 0.259·35-s − 1.98·37-s + 1.55·41-s + 1.74·43-s + 0.338·47-s − 0.662·49-s − 0.0205·53-s − 0.116·55-s + 1.43·59-s + 0.567·61-s − 0.0172·65-s − 1.28·67-s − 0.867·71-s + 0.833·73-s + 0.150·77-s + 0.764·79-s − 1.16·83-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

\(L(1)\) \(\approx\) \(1.877991626\)
\(L(\frac12)\) \(\approx\) \(1.877991626\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 - 0.860T + 11T^{2} \)
13 \( 1 - 0.139T + 13T^{2} \)
17 \( 1 + 5.50T + 17T^{2} \)
19 \( 1 - 5.25T + 19T^{2} \)
29 \( 1 + 9.76T + 29T^{2} \)
31 \( 1 - 6.78T + 31T^{2} \)
37 \( 1 + 12.0T + 37T^{2} \)
41 \( 1 - 9.98T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 2.32T + 47T^{2} \)
53 \( 1 + 0.149T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 4.43T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + 7.31T + 71T^{2} \)
73 \( 1 - 7.11T + 73T^{2} \)
79 \( 1 - 6.79T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 1.93T + 97T^{2} \)
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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

−7.62972022752656823369229699301, −7.31167015427148941395512174424, −6.48875422682276677752665038972, −5.66912112607525873254510319463, −4.98473578067983757012324953544, −4.24745438364276635843989537102, −3.61417851928508586607543127149, −2.63118188237224067983570784240, −1.75996223251628944686473893423, −0.68180340617725493237626046191, 0.68180340617725493237626046191, 1.75996223251628944686473893423, 2.63118188237224067983570784240, 3.61417851928508586607543127149, 4.24745438364276635843989537102, 4.98473578067983757012324953544, 5.66912112607525873254510319463, 6.48875422682276677752665038972, 7.31167015427148941395512174424, 7.62972022752656823369229699301

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