Properties

Label 2-2646-1.1-c1-0-30
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 $0$

Origins

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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: $\chi_{2646} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.684731199\)
\(L(\frac12)\) \(\approx\) \(3.684731199\)
\(L(\frac{3}{2})\) 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 \( 1 - 14.8T + 97T^{2} \)
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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

−8.904402627740010420481192853669, −8.012897113797102669409786590718, −7.17410808353620848711608687012, −6.43660470973841071689009844649, −5.58889081562060924951492103563, −5.21895273110017556064096732903, −3.82454224164207829957955491269, −3.45810404312383599362920006109, −2.12787860985199687341077284804, −1.20995913547507016240539929411, 1.20995913547507016240539929411, 2.12787860985199687341077284804, 3.45810404312383599362920006109, 3.82454224164207829957955491269, 5.21895273110017556064096732903, 5.58889081562060924951492103563, 6.43660470973841071689009844649, 7.17410808353620848711608687012, 8.012897113797102669409786590718, 8.904402627740010420481192853669

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