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Properties

Label 2-84e2-1.1-c1-0-82
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·13-s − 7·19-s − 5·25-s − 7·31-s − 37-s − 5·43-s − 14·61-s − 11·67-s + 7·73-s + 13·79-s − 14·97-s − 7·103-s + 17·109-s + ⋯
L(s)  = 1  + 1.94·13-s − 1.60·19-s − 25-s − 1.25·31-s − 0.164·37-s − 0.762·43-s − 1.79·61-s − 1.34·67-s + 0.819·73-s + 1.46·79-s − 1.42·97-s − 0.689·103-s + 1.62·109-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: $-1$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.69815649441562162377076845186, −6.76539086791331015285768232597, −6.16030207005109348850349064165, −5.68006317530212705372525394749, −4.63225465371620530951751446236, −3.88869221547929279438653829739, −3.34454675279272923232386773281, −2.13037435943249941765801940949, −1.39324048097838607810358577106, 0, 1.39324048097838607810358577106, 2.13037435943249941765801940949, 3.34454675279272923232386773281, 3.88869221547929279438653829739, 4.63225465371620530951751446236, 5.68006317530212705372525394749, 6.16030207005109348850349064165, 6.76539086791331015285768232597, 7.69815649441562162377076845186

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