Properties

Label 2-2646-1.1-c1-0-48
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s + 2.24·11-s − 3·13-s + 16-s − 4.41·17-s − 4.58·19-s − 1.41·20-s + 2.24·22-s − 23-s − 2.99·25-s − 3·26-s − 5.24·29-s + 1.24·31-s + 32-s − 4.41·34-s − 10.4·37-s − 4.58·38-s − 1.41·40-s + 2.82·41-s + 3.24·43-s + 2.24·44-s − 46-s + 7.07·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 0.447·10-s + 0.676·11-s − 0.832·13-s + 0.250·16-s − 1.07·17-s − 1.05·19-s − 0.316·20-s + 0.478·22-s − 0.208·23-s − 0.599·25-s − 0.588·26-s − 0.973·29-s + 0.223·31-s + 0.176·32-s − 0.757·34-s − 1.72·37-s − 0.743·38-s − 0.223·40-s + 0.441·41-s + 0.494·43-s + 0.338·44-s − 0.147·46-s + 1.03·47-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 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 + 4.58T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 5.24T + 29T^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 0.171T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 5.48T + 71T^{2} \)
73 \( 1 + 9.89T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 - 1.92T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.500793991931724638653489871972, −7.46735537077015873032142628226, −6.99050028109064832252254631565, −6.13500277211491840435553766669, −5.30192695990541561140331364463, −4.21616604767026033961272550955, −3.99537247150537430840121571645, −2.71560876310166327303284802940, −1.80220743335325392270577860498, 0, 1.80220743335325392270577860498, 2.71560876310166327303284802940, 3.99537247150537430840121571645, 4.21616604767026033961272550955, 5.30192695990541561140331364463, 6.13500277211491840435553766669, 6.99050028109064832252254631565, 7.46735537077015873032142628226, 8.500793991931724638653489871972

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