Properties

Label 2-84e2-1.1-c1-0-45
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  + 3·5-s + 3·11-s + 4·13-s − 4·19-s + 4·25-s − 9·29-s − 31-s + 8·37-s + 10·43-s + 6·47-s + 3·53-s + 9·55-s − 3·59-s + 10·61-s + 12·65-s + 10·67-s − 6·71-s − 2·73-s + 79-s + 9·83-s + 6·89-s − 12·95-s + 97-s − 18·101-s + 8·103-s − 3·107-s + 14·109-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.904·11-s + 1.10·13-s − 0.917·19-s + 4/5·25-s − 1.67·29-s − 0.179·31-s + 1.31·37-s + 1.52·43-s + 0.875·47-s + 0.412·53-s + 1.21·55-s − 0.390·59-s + 1.28·61-s + 1.48·65-s + 1.22·67-s − 0.712·71-s − 0.234·73-s + 0.112·79-s + 0.987·83-s + 0.635·89-s − 1.23·95-s + 0.101·97-s − 1.79·101-s + 0.788·103-s − 0.290·107-s + 1.34·109-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: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.091382749\)
\(L(\frac12)\) \(\approx\) \(3.091382749\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−7.973598604422608856711125899156, −7.12684258637692642405801706265, −6.30981704578337725497607593309, −5.97708585339505464195745997258, −5.33778120225130226806025517412, −4.21075854793680523594236828144, −3.71168889085604917791465076443, −2.50214901196819952049505879216, −1.83818484293011714855169652362, −0.942551933080821992746372320770, 0.942551933080821992746372320770, 1.83818484293011714855169652362, 2.50214901196819952049505879216, 3.71168889085604917791465076443, 4.21075854793680523594236828144, 5.33778120225130226806025517412, 5.97708585339505464195745997258, 6.30981704578337725497607593309, 7.12684258637692642405801706265, 7.973598604422608856711125899156

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