Properties

Label 2-84e2-1.1-c1-0-35
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  + 4·13-s + 6·17-s + 2·19-s − 5·25-s + 6·29-s − 4·31-s + 2·37-s + 6·41-s − 8·43-s + 12·47-s − 6·53-s + 6·59-s − 8·61-s + 4·67-s − 2·73-s − 8·79-s + 6·83-s − 6·89-s + 10·97-s − 4·103-s + 12·107-s + 2·109-s − 6·113-s + ⋯
L(s)  = 1  + 1.10·13-s + 1.45·17-s + 0.458·19-s − 25-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s − 0.824·53-s + 0.781·59-s − 1.02·61-s + 0.488·67-s − 0.234·73-s − 0.900·79-s + 0.658·83-s − 0.635·89-s + 1.01·97-s − 0.394·103-s + 1.16·107-s + 0.191·109-s − 0.564·113-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

\(L(1)\) \(\approx\) \(2.313967214\)
\(L(\frac12)\) \(\approx\) \(2.313967214\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−7.88115201382147028507134927026, −7.39293674981761532961441986375, −6.43902889849944115155691598253, −5.83259812621372959122100845675, −5.25590580602395553760699468065, −4.24465289176609138955904372614, −3.56550924639142318669266713404, −2.84501304641458537647904209821, −1.67921161671410140538744564928, −0.818365395076437722978448101852, 0.818365395076437722978448101852, 1.67921161671410140538744564928, 2.84501304641458537647904209821, 3.56550924639142318669266713404, 4.24465289176609138955904372614, 5.25590580602395553760699468065, 5.83259812621372959122100845675, 6.43902889849944115155691598253, 7.39293674981761532961441986375, 7.88115201382147028507134927026

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