Properties

Label 2-84e2-1.1-c1-0-47
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  + 5-s + 3·11-s + 4·13-s + 4·19-s + 8·23-s − 4·25-s + 3·29-s + 5·31-s + 8·37-s − 8·41-s − 6·43-s + 10·47-s − 9·53-s + 3·55-s − 5·59-s − 10·61-s + 4·65-s − 6·67-s + 10·71-s + 2·73-s − 11·79-s + 7·83-s + 18·89-s + 4·95-s − 17·97-s + 2·101-s − 11·107-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.904·11-s + 1.10·13-s + 0.917·19-s + 1.66·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 1.31·37-s − 1.24·41-s − 0.914·43-s + 1.45·47-s − 1.23·53-s + 0.404·55-s − 0.650·59-s − 1.28·61-s + 0.496·65-s − 0.733·67-s + 1.18·71-s + 0.234·73-s − 1.23·79-s + 0.768·83-s + 1.90·89-s + 0.410·95-s − 1.72·97-s + 0.199·101-s − 1.06·107-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: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.837167303\)
\(L(\frac12)\) \(\approx\) \(2.837167303\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−7.998157978762509709012849864254, −7.16183400955335543876118219905, −6.44162281874474764302538262339, −5.99193522513282755009195665126, −5.11162449121302522698816434748, −4.38619854192959487427000112591, −3.47925632125466026069668696822, −2.84811242361592356592684674685, −1.61004889127807801694244292086, −0.948672759887688732959170989133, 0.948672759887688732959170989133, 1.61004889127807801694244292086, 2.84811242361592356592684674685, 3.47925632125466026069668696822, 4.38619854192959487427000112591, 5.11162449121302522698816434748, 5.99193522513282755009195665126, 6.44162281874474764302538262339, 7.16183400955335543876118219905, 7.998157978762509709012849864254

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