Properties

Degree $2$
Conductor $7056$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 4·11-s − 4·13-s + 4·19-s + 11·25-s − 2·29-s + 8·31-s − 6·37-s − 4·43-s + 8·47-s + 10·53-s − 16·55-s − 4·59-s + 4·61-s − 16·65-s − 4·67-s + 8·71-s + 16·73-s + 8·79-s + 12·83-s + 8·89-s + 16·95-s − 8·97-s + 4·101-s − 8·103-s + 4·107-s − 14·109-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.20·11-s − 1.10·13-s + 0.917·19-s + 11/5·25-s − 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.609·43-s + 1.16·47-s + 1.37·53-s − 2.15·55-s − 0.520·59-s + 0.512·61-s − 1.98·65-s − 0.488·67-s + 0.949·71-s + 1.87·73-s + 0.900·79-s + 1.31·83-s + 0.847·89-s + 1.64·95-s − 0.812·97-s + 0.398·101-s − 0.788·103-s + 0.386·107-s − 1.34·109-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: $1$
Motivic weight: \(1\)
Character: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.656413816\)
\(L(\frac12)\) \(\approx\) \(2.656413816\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.916680638172122349209180978402, −7.15626924228060006589866403814, −6.53223269447526142619685193559, −5.64836882923047051143323621301, −5.26001411946587964391418777104, −4.69336426412013614248722189397, −3.32288223921369618581771511826, −2.48298505359813833416816457120, −2.06015196877250864953474557133, −0.818745509716799952266967219226, 0.818745509716799952266967219226, 2.06015196877250864953474557133, 2.48298505359813833416816457120, 3.32288223921369618581771511826, 4.69336426412013614248722189397, 5.26001411946587964391418777104, 5.64836882923047051143323621301, 6.53223269447526142619685193559, 7.15626924228060006589866403814, 7.916680638172122349209180978402

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