Properties

Label 2-2646-1.1-c1-0-35
Degree $2$
Conductor $2646$
Sign $-1$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 2·13-s + 16-s − 6·17-s + 5·19-s − 20-s + 22-s + 3·23-s − 4·25-s − 2·26-s + 2·29-s − 5·31-s − 32-s + 6·34-s + 3·37-s − 5·38-s + 40-s + 3·41-s − 2·43-s − 44-s − 3·46-s − 10·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 1.14·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s − 4/5·25-s − 0.392·26-s + 0.371·29-s − 0.898·31-s − 0.176·32-s + 1.02·34-s + 0.493·37-s − 0.811·38-s + 0.158·40-s + 0.468·41-s − 0.304·43-s − 0.150·44-s − 0.442·46-s − 1.45·47-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: yes
Arithmetic: yes
Character: $\chi_{2646} (1, \cdot )$
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 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.551423168398543236619112644628, −7.75240721792102384696135904334, −7.14137669295085940317588036997, −6.34002059224990781957978985695, −5.45883024829548815631361397772, −4.46934491133839587358262773639, −3.50899548863681658313549736296, −2.55069351015900942552710835973, −1.37982935479584858529094782557, 0, 1.37982935479584858529094782557, 2.55069351015900942552710835973, 3.50899548863681658313549736296, 4.46934491133839587358262773639, 5.45883024829548815631361397772, 6.34002059224990781957978985695, 7.14137669295085940317588036997, 7.75240721792102384696135904334, 8.551423168398543236619112644628

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