Properties

Label 2-2646-1.1-c1-0-12
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 − 6.24·11-s + 3·13-s + 16-s + 1.58·17-s + 7.41·19-s − 1.41·20-s − 6.24·22-s − 23-s − 2.99·25-s + 3·26-s + 3.24·29-s + 7.24·31-s + 32-s + 1.58·34-s + 6.48·37-s + 7.41·38-s − 1.41·40-s + 2.82·41-s − 5.24·43-s − 6.24·44-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 − 1.88·11-s + 0.832·13-s + 0.250·16-s + 0.384·17-s + 1.70·19-s − 0.316·20-s − 1.33·22-s − 0.208·23-s − 0.599·25-s + 0.588·26-s + 0.602·29-s + 1.30·31-s + 0.176·32-s + 0.271·34-s + 1.06·37-s + 1.20·38-s − 0.223·40-s + 0.441·41-s − 0.799·43-s − 0.941·44-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

\(L(1)\) \(\approx\) \(2.448592657\)
\(L(\frac12)\) \(\approx\) \(2.448592657\)
\(L(\frac{3}{2})\) 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 \( 1 + 6.24T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 - 1.58T + 17T^{2} \)
19 \( 1 - 7.41T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 - 7.24T + 31T^{2} \)
37 \( 1 - 6.48T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 - 2.75T + 53T^{2} \)
59 \( 1 - 5.82T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 - 4.75T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 9.89T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + 5.31T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 9.17T + 97T^{2} \)
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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

−8.622644129773862539529832504706, −7.79379376956131483533436271788, −7.59407855485482138496048055403, −6.43294557206771289998648121994, −5.58091837216949866736706019275, −5.02328257322575875980958178394, −4.05524389291606364540711530746, −3.19336032846081947983380085263, −2.46674086760496643756866727522, −0.890384696189987934251088519419, 0.890384696189987934251088519419, 2.46674086760496643756866727522, 3.19336032846081947983380085263, 4.05524389291606364540711530746, 5.02328257322575875980958178394, 5.58091837216949866736706019275, 6.43294557206771289998648121994, 7.59407855485482138496048055403, 7.79379376956131483533436271788, 8.622644129773862539529832504706

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