Properties

Label 2-6840-1.1-c1-0-53
Degree $2$
Conductor $6840$
Sign $-1$
Analytic cond. $54.6176$
Root an. cond. $7.39037$
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-s − 4·11-s + 2·13-s − 6·17-s − 19-s + 4·23-s + 25-s + 10·29-s + 4·31-s + 10·37-s − 6·41-s + 12·47-s − 7·49-s − 10·53-s + 4·55-s − 10·61-s − 2·65-s + 4·67-s + 10·73-s + 4·79-s − 12·83-s + 6·85-s − 14·89-s + 95-s − 2·97-s − 6·101-s − 16·103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 1.64·37-s − 0.937·41-s + 1.75·47-s − 49-s − 1.37·53-s + 0.539·55-s − 1.28·61-s − 0.248·65-s + 0.488·67-s + 1.17·73-s + 0.450·79-s − 1.31·83-s + 0.650·85-s − 1.48·89-s + 0.102·95-s − 0.203·97-s − 0.597·101-s − 1.57·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6840\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(54.6176\)
Root analytic conductor: \(7.39037\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6840,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
19 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−7.81289355199641979551370165165, −6.75455879245676649787874977741, −6.43692522238243133727003650056, −5.40670078441511981591888331586, −4.65912254521612704782286522168, −4.16377166035292820952539672011, −2.95681836686443709979247590489, −2.53312602620037045170501575440, −1.19074048156047241483019349201, 0, 1.19074048156047241483019349201, 2.53312602620037045170501575440, 2.95681836686443709979247590489, 4.16377166035292820952539672011, 4.65912254521612704782286522168, 5.40670078441511981591888331586, 6.43692522238243133727003650056, 6.75455879245676649787874977741, 7.81289355199641979551370165165

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