Properties

Label 2-8712-1.1-c1-0-62
Degree $2$
Conductor $8712$
Sign $1$
Analytic cond. $69.5656$
Root an. cond. $8.34060$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·13-s + 2·17-s + 4·19-s + 8·23-s − 25-s + 6·29-s + 8·31-s + 6·37-s − 6·41-s − 4·43-s − 7·49-s + 2·53-s − 4·59-s + 2·61-s + 4·65-s − 4·67-s − 8·71-s − 10·73-s + 8·79-s − 4·83-s + 4·85-s + 6·89-s + 8·95-s + 2·97-s − 18·101-s + 16·103-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s − 49-s + 0.274·53-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.439·83-s + 0.433·85-s + 0.635·89-s + 0.820·95-s + 0.203·97-s − 1.79·101-s + 1.57·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8712\)    =    \(2^{3} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(69.5656\)
Root analytic conductor: \(8.34060\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8712} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8712,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.003209511\)
\(L(\frac12)\) \(\approx\) \(3.003209511\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.83654723068629114933839597949, −6.92796334648786182474479663711, −6.40725479078387541676599821855, −5.70681982833013824339552087978, −5.05736853539339825666279063330, −4.38461660538881471089251752313, −3.21029387756548432515894957212, −2.80691305312846013255691010566, −1.61011393533291281462985460586, −0.918192672118257559483050015003, 0.918192672118257559483050015003, 1.61011393533291281462985460586, 2.80691305312846013255691010566, 3.21029387756548432515894957212, 4.38461660538881471089251752313, 5.05736853539339825666279063330, 5.70681982833013824339552087978, 6.40725479078387541676599821855, 6.92796334648786182474479663711, 7.83654723068629114933839597949

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