Properties

Label 2-8712-1.1-c1-0-122
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 6·13-s − 6·17-s + 2·19-s − 8·23-s − 5·25-s + 2·29-s − 4·31-s + 2·37-s − 10·41-s + 6·43-s + 4·47-s − 3·49-s − 4·53-s − 4·59-s + 2·61-s − 8·67-s − 12·71-s + 2·73-s − 14·79-s − 4·83-s + 12·91-s + 2·97-s − 14·101-s − 16·103-s + 4·107-s − 10·109-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.66·13-s − 1.45·17-s + 0.458·19-s − 1.66·23-s − 25-s + 0.371·29-s − 0.718·31-s + 0.328·37-s − 1.56·41-s + 0.914·43-s + 0.583·47-s − 3/7·49-s − 0.549·53-s − 0.520·59-s + 0.256·61-s − 0.977·67-s − 1.42·71-s + 0.234·73-s − 1.57·79-s − 0.439·83-s + 1.25·91-s + 0.203·97-s − 1.39·101-s − 1.57·103-s + 0.386·107-s − 0.957·109-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8712,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−7.53132753837424459565387809235, −6.65538524121335935736951890557, −6.01624298500087106098727240501, −5.48883417736729230439987281816, −4.38863218231925665898837742520, −4.06649425401559734263429239216, −3.11557930476375974767686690063, −1.99948079163893685970463261538, −1.43553598633874759690956925011, 0, 1.43553598633874759690956925011, 1.99948079163893685970463261538, 3.11557930476375974767686690063, 4.06649425401559734263429239216, 4.38863218231925665898837742520, 5.48883417736729230439987281816, 6.01624298500087106098727240501, 6.65538524121335935736951890557, 7.53132753837424459565387809235

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