Properties

Label 2-6900-1.1-c1-0-53
Degree $2$
Conductor $6900$
Sign $-1$
Analytic cond. $55.0967$
Root an. cond. $7.42272$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.22·7-s + 9-s + 2.15·11-s − 3.08·13-s + 5.11·17-s − 1.80·19-s − 3.22·21-s − 23-s + 27-s + 7.18·29-s − 1.50·31-s + 2.15·33-s − 11.1·37-s − 3.08·39-s − 6.38·41-s − 5.16·43-s − 1.19·47-s + 3.42·49-s + 5.11·51-s − 2.57·53-s − 1.80·57-s + 9.34·59-s − 7.96·61-s − 3.22·63-s − 5.30·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.22·7-s + 0.333·9-s + 0.650·11-s − 0.855·13-s + 1.24·17-s − 0.414·19-s − 0.704·21-s − 0.208·23-s + 0.192·27-s + 1.33·29-s − 0.270·31-s + 0.375·33-s − 1.82·37-s − 0.493·39-s − 0.997·41-s − 0.788·43-s − 0.174·47-s + 0.489·49-s + 0.716·51-s − 0.353·53-s − 0.239·57-s + 1.21·59-s − 1.01·61-s − 0.406·63-s − 0.647·67-s − 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(55.0967\)
Root analytic conductor: \(7.42272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6900,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
23 \( 1 + T \)
good7 \( 1 + 3.22T + 7T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + 3.08T + 13T^{2} \)
17 \( 1 - 5.11T + 17T^{2} \)
19 \( 1 + 1.80T + 19T^{2} \)
29 \( 1 - 7.18T + 29T^{2} \)
31 \( 1 + 1.50T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 6.38T + 41T^{2} \)
43 \( 1 + 5.16T + 43T^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 + 2.57T + 53T^{2} \)
59 \( 1 - 9.34T + 59T^{2} \)
61 \( 1 + 7.96T + 61T^{2} \)
67 \( 1 + 5.30T + 67T^{2} \)
71 \( 1 + 0.966T + 71T^{2} \)
73 \( 1 - 4.23T + 73T^{2} \)
79 \( 1 - 7.53T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 9.81T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−7.63177221135967809868526999298, −6.75162660074541119869908037037, −6.52278111847377456887774115265, −5.45848710483732358047988122948, −4.74037169221083835702505383475, −3.66886950535998805023591968210, −3.29707178007515555259287788947, −2.41235465497347376208558004838, −1.35711321392151637092765759526, 0, 1.35711321392151637092765759526, 2.41235465497347376208558004838, 3.29707178007515555259287788947, 3.66886950535998805023591968210, 4.74037169221083835702505383475, 5.45848710483732358047988122948, 6.52278111847377456887774115265, 6.75162660074541119869908037037, 7.63177221135967809868526999298

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