Properties

Label 2-84e2-1.1-c1-0-51
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  − 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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
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

−7.72337326384857641029833442753, −6.83938821823237049403422486214, −6.55243883519486721499599903047, −5.18410365609541501678857709232, −4.70141834001079247284550570560, −3.97255156925627183868646785583, −3.25266724848457980559188678742, −2.47482686238535867799638700086, −1.02250477174083311248738777220, 0, 1.02250477174083311248738777220, 2.47482686238535867799638700086, 3.25266724848457980559188678742, 3.97255156925627183868646785583, 4.70141834001079247284550570560, 5.18410365609541501678857709232, 6.55243883519486721499599903047, 6.83938821823237049403422486214, 7.72337326384857641029833442753

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