Properties

Label 2-84e2-1.1-c1-0-92
Degree $2$
Conductor $7056$
Sign $-1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.46·5-s − 3.46·11-s − 2·13-s − 3.46·17-s − 4·19-s + 3.46·23-s + 6.99·25-s − 4·31-s + 2·37-s − 10.3·41-s + 4·43-s + 6.92·47-s − 6.92·53-s − 11.9·55-s − 6.92·59-s + 10·61-s − 6.92·65-s + 4·67-s + 10.3·71-s − 14·73-s − 8·79-s − 11.9·85-s + 3.46·89-s − 13.8·95-s − 14·97-s − 3.46·101-s − 4·103-s + ⋯
L(s)  = 1  + 1.54·5-s − 1.04·11-s − 0.554·13-s − 0.840·17-s − 0.917·19-s + 0.722·23-s + 1.39·25-s − 0.718·31-s + 0.328·37-s − 1.62·41-s + 0.609·43-s + 1.01·47-s − 0.951·53-s − 1.61·55-s − 0.901·59-s + 1.28·61-s − 0.859·65-s + 0.488·67-s + 1.23·71-s − 1.63·73-s − 0.900·79-s − 1.30·85-s + 0.367·89-s − 1.42·95-s − 1.42·97-s − 0.344·101-s − 0.394·103-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: no
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 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−7.51218116529908702620620166504, −6.77167896444280890855160561782, −6.21512305737288936287229642991, −5.37006982049676306500307958175, −5.02114905098027128279915958922, −4.05548690470664349380117502531, −2.79489885101287138119063207250, −2.34188766022526201091472767243, −1.50682940868252350405565775611, 0, 1.50682940868252350405565775611, 2.34188766022526201091472767243, 2.79489885101287138119063207250, 4.05548690470664349380117502531, 5.02114905098027128279915958922, 5.37006982049676306500307958175, 6.21512305737288936287229642991, 6.77167896444280890855160561782, 7.51218116529908702620620166504

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