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Properties

Label 2-84e2-1.1-c1-0-80
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  + 4·13-s − 4·17-s + 4·19-s + 4·23-s − 5·25-s − 2·29-s − 8·31-s − 6·37-s − 12·41-s − 4·43-s − 8·47-s − 6·53-s + 12·59-s + 4·61-s + 4·67-s − 12·71-s + 8·73-s + 16·79-s − 4·83-s − 4·89-s + 16·97-s + 8·101-s + 8·103-s + 8·107-s − 14·109-s − 2·113-s + ⋯
L(s)  = 1  + 1.10·13-s − 0.970·17-s + 0.917·19-s + 0.834·23-s − 25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 1.87·41-s − 0.609·43-s − 1.16·47-s − 0.824·53-s + 1.56·59-s + 0.512·61-s + 0.488·67-s − 1.42·71-s + 0.936·73-s + 1.80·79-s − 0.439·83-s − 0.423·89-s + 1.62·97-s + 0.796·101-s + 0.788·103-s + 0.773·107-s − 1.34·109-s − 0.188·113-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: 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−7.55368824593198046193072046178, −6.85773872648419481689700481552, −6.28934640908204385833239215572, −5.37056011209214751870181490050, −4.90597207574689985720237337950, −3.68113539814298466424469342291, −3.44618650812695564260584942391, −2.14771171104644400460077223399, −1.37724144475974014148789786179, 0, 1.37724144475974014148789786179, 2.14771171104644400460077223399, 3.44618650812695564260584942391, 3.68113539814298466424469342291, 4.90597207574689985720237337950, 5.37056011209214751870181490050, 6.28934640908204385833239215572, 6.85773872648419481689700481552, 7.55368824593198046193072046178

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