Properties

Label 2-84e2-1.1-c1-0-72
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  − 5·13-s − 19-s − 5·25-s + 11·31-s + 11·37-s + 13·43-s − 14·61-s − 5·67-s − 17·73-s − 17·79-s − 14·97-s − 13·103-s − 19·109-s + ⋯
L(s)  = 1  − 1.38·13-s − 0.229·19-s − 25-s + 1.97·31-s + 1.80·37-s + 1.98·43-s − 1.79·61-s − 0.610·67-s − 1.98·73-s − 1.91·79-s − 1.42·97-s − 1.28·103-s − 1.81·109-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: $\chi_{7056} (1, \cdot )$
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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
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

−7.67306718287486929393594560851, −6.96827923611881829072103831107, −6.11112759203424531180371812898, −5.59746425303006995358444442730, −4.43289275430421790726586679842, −4.33631159443367309398800515424, −2.89472948302706746664620893750, −2.47752709868336174623609828675, −1.26055540851728166156733522039, 0, 1.26055540851728166156733522039, 2.47752709868336174623609828675, 2.89472948302706746664620893750, 4.33631159443367309398800515424, 4.43289275430421790726586679842, 5.59746425303006995358444442730, 6.11112759203424531180371812898, 6.96827923611881829072103831107, 7.67306718287486929393594560851

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