Properties

Label 2-84e2-1.1-c1-0-13
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  − 4·5-s + 3·13-s + 4·17-s + 7·19-s − 4·23-s + 11·25-s − 8·29-s − 5·31-s + 3·37-s − 8·41-s − 11·43-s + 4·47-s + 4·53-s + 12·59-s + 2·61-s − 12·65-s + 3·67-s − 12·71-s − 73-s − 79-s + 12·83-s − 16·85-s − 8·89-s − 28·95-s + 2·97-s + 3·103-s + 12·107-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.832·13-s + 0.970·17-s + 1.60·19-s − 0.834·23-s + 11/5·25-s − 1.48·29-s − 0.898·31-s + 0.493·37-s − 1.24·41-s − 1.67·43-s + 0.583·47-s + 0.549·53-s + 1.56·59-s + 0.256·61-s − 1.48·65-s + 0.366·67-s − 1.42·71-s − 0.117·73-s − 0.112·79-s + 1.31·83-s − 1.73·85-s − 0.847·89-s − 2.87·95-s + 0.203·97-s + 0.295·103-s + 1.16·107-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

\(L(1)\) \(\approx\) \(1.198241711\)
\(L(\frac12)\) \(\approx\) \(1.198241711\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.902411121628635971953033958212, −7.37786121519322206420381337939, −6.79574495224184795847290893333, −5.66563702886740437379671841736, −5.17520107670724123245452546978, −4.08173413639678390137451260087, −3.61646063426785494401781033005, −3.10595913766941266781565341125, −1.62894710985601508842847294311, −0.57426866169805545305079539621, 0.57426866169805545305079539621, 1.62894710985601508842847294311, 3.10595913766941266781565341125, 3.61646063426785494401781033005, 4.08173413639678390137451260087, 5.17520107670724123245452546978, 5.66563702886740437379671841736, 6.79574495224184795847290893333, 7.37786121519322206420381337939, 7.902411121628635971953033958212

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