Properties

Label 2-84e2-1.1-c1-0-30
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 + 3·13-s + 8·17-s − 19-s + 8·23-s − 25-s − 4·29-s + 3·31-s − 37-s + 6·41-s − 11·43-s − 6·47-s + 12·53-s − 4·55-s − 4·59-s + 6·61-s − 6·65-s − 13·67-s − 10·71-s + 11·73-s + 3·79-s − 2·83-s − 16·85-s + 2·95-s − 10·97-s + 10·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.603·11-s + 0.832·13-s + 1.94·17-s − 0.229·19-s + 1.66·23-s − 1/5·25-s − 0.742·29-s + 0.538·31-s − 0.164·37-s + 0.937·41-s − 1.67·43-s − 0.875·47-s + 1.64·53-s − 0.539·55-s − 0.520·59-s + 0.768·61-s − 0.744·65-s − 1.58·67-s − 1.18·71-s + 1.28·73-s + 0.337·79-s − 0.219·83-s − 1.73·85-s + 0.205·95-s − 1.01·97-s + 0.995·101-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: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.962042379\)
\(L(\frac12)\) \(\approx\) \(1.962042379\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−7.902948231459067302935552346558, −7.33475219528809330872600366668, −6.62591164060096215810738704248, −5.81204583549235331357934123304, −5.13277720066640461862378462543, −4.21939064268804023339027371565, −3.53018349026108514725570576258, −3.02958828511922128937801832292, −1.59275250042016111667743471768, −0.76499152982008805008093192117, 0.76499152982008805008093192117, 1.59275250042016111667743471768, 3.02958828511922128937801832292, 3.53018349026108514725570576258, 4.21939064268804023339027371565, 5.13277720066640461862378462543, 5.81204583549235331357934123304, 6.62591164060096215810738704248, 7.33475219528809330872600366668, 7.902948231459067302935552346558

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