Properties

Label 2-2646-1.1-c1-0-22
Degree $2$
Conductor $2646$
Sign $-1$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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  − 2-s + 4-s − 4.41·5-s − 8-s + 4.41·10-s − 11-s − 4.24·13-s + 16-s + 2.82·17-s + 5.82·19-s − 4.41·20-s + 22-s + 5.24·23-s + 14.4·25-s + 4.24·26-s − 0.242·29-s + 7.24·31-s − 32-s − 2.82·34-s + 5.24·37-s − 5.82·38-s + 4.41·40-s − 6.17·41-s − 6.48·43-s − 44-s − 5.24·46-s − 11.6·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.97·5-s − 0.353·8-s + 1.39·10-s − 0.301·11-s − 1.17·13-s + 0.250·16-s + 0.685·17-s + 1.33·19-s − 0.987·20-s + 0.213·22-s + 1.09·23-s + 2.89·25-s + 0.832·26-s − 0.0450·29-s + 1.30·31-s − 0.176·32-s − 0.485·34-s + 0.861·37-s − 0.945·38-s + 0.697·40-s − 0.963·41-s − 0.988·43-s − 0.150·44-s − 0.772·46-s − 1.70·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4.41T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 - 5.24T + 23T^{2} \)
29 \( 1 + 0.242T + 29T^{2} \)
31 \( 1 - 7.24T + 31T^{2} \)
37 \( 1 - 5.24T + 37T^{2} \)
41 \( 1 + 6.17T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 1.75T + 67T^{2} \)
71 \( 1 + 1.24T + 71T^{2} \)
73 \( 1 - 3.17T + 73T^{2} \)
79 \( 1 + 6.48T + 79T^{2} \)
83 \( 1 - 3.89T + 83T^{2} \)
89 \( 1 + 3.34T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−8.286792238368949617331425222827, −7.70760115156622673825612550338, −7.34433610062321642573310067817, −6.53300645337732712080617026304, −5.12045544713022865077138295735, −4.60330205049140923825091341255, −3.32796979722954235934600701890, −2.90954569201735560829534542417, −1.12483444901762346494863767768, 0, 1.12483444901762346494863767768, 2.90954569201735560829534542417, 3.32796979722954235934600701890, 4.60330205049140923825091341255, 5.12045544713022865077138295735, 6.53300645337732712080617026304, 7.34433610062321642573310067817, 7.70760115156622673825612550338, 8.286792238368949617331425222827

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