Properties

Label 2-84e2-1.1-c1-0-83
Degree $2$
Conductor $7056$
Sign $-1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  + 2·5-s − 6·13-s − 2·17-s + 4·19-s − 4·23-s − 25-s + 10·29-s − 8·31-s + 6·37-s − 2·41-s + 4·43-s − 8·47-s + 10·53-s − 12·59-s + 2·61-s − 12·65-s − 12·67-s − 12·71-s + 14·73-s + 8·79-s − 12·83-s − 4·85-s − 2·89-s + 8·95-s − 10·97-s − 6·101-s − 2·109-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.66·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.85·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1.37·53-s − 1.56·59-s + 0.256·61-s − 1.48·65-s − 1.46·67-s − 1.42·71-s + 1.63·73-s + 0.900·79-s − 1.31·83-s − 0.433·85-s − 0.211·89-s + 0.820·95-s − 1.01·97-s − 0.597·101-s − 0.191·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.56653311845079447226607769195, −6.89668090429251782030846038586, −6.16314397903413349396524741780, −5.46156276712624625792211688667, −4.84472447638438006928239933663, −4.07712068075749994728906530737, −2.90297487998668631853198734501, −2.34235955836176630368170451259, −1.41502771567863508498611693583, 0, 1.41502771567863508498611693583, 2.34235955836176630368170451259, 2.90297487998668631853198734501, 4.07712068075749994728906530737, 4.84472447638438006928239933663, 5.46156276712624625792211688667, 6.16314397903413349396524741780, 6.89668090429251782030846038586, 7.56653311845079447226607769195

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