Properties

Label 2-84e2-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·11-s − 2·13-s − 6·17-s − 4·19-s − 6·23-s − 25-s − 4·31-s + 10·37-s − 2·41-s + 4·43-s + 4·47-s − 12·53-s + 4·55-s + 12·59-s − 6·61-s + 4·65-s + 4·67-s + 14·71-s + 2·73-s + 8·79-s − 16·83-s + 12·85-s + 6·89-s + 8·95-s + 18·97-s − 14·101-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.603·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 1/5·25-s − 0.718·31-s + 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.583·47-s − 1.64·53-s + 0.539·55-s + 1.56·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s + 1.66·71-s + 0.234·73-s + 0.900·79-s − 1.75·83-s + 1.30·85-s + 0.635·89-s + 0.820·95-s + 1.82·97-s − 1.39·101-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6419445198\)
\(L(\frac12)\) \(\approx\) \(0.6419445198\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−7.971035293400414999497017290309, −7.35982312744619529123795696792, −6.57020783298166711015809988104, −5.92435180370351418708688459740, −4.95895066506278925315287215220, −4.27294605202891169369750413399, −3.77748816967912386387626041524, −2.59734463484332742401745770212, −2.00735733350817887845048818343, −0.38154394322912952085545456289, 0.38154394322912952085545456289, 2.00735733350817887845048818343, 2.59734463484332742401745770212, 3.77748816967912386387626041524, 4.27294605202891169369750413399, 4.95895066506278925315287215220, 5.92435180370351418708688459740, 6.57020783298166711015809988104, 7.35982312744619529123795696792, 7.971035293400414999497017290309

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