Properties

Label 2-2646-1.1-c1-0-38
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 − 8-s − 5·13-s + 16-s + 3·17-s − 2·19-s + 9·23-s − 5·25-s + 5·26-s + 3·29-s − 5·31-s − 32-s − 3·34-s + 2·37-s + 2·38-s − 6·41-s − 43-s − 9·46-s − 6·47-s + 5·50-s − 5·52-s − 3·53-s − 3·58-s − 3·59-s + 10·61-s + 5·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.38·13-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 1.87·23-s − 25-s + 0.980·26-s + 0.557·29-s − 0.898·31-s − 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s − 0.152·43-s − 1.32·46-s − 0.875·47-s + 0.707·50-s − 0.693·52-s − 0.412·53-s − 0.393·58-s − 0.390·59-s + 1.28·61-s + 0.635·62-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.574049298258472467392444731185, −7.62887873867370361852526319092, −7.21285497786280286801180495079, −6.33738862277121156447234649392, −5.36650684226892599309211492342, −4.64577805751899849665319383092, −3.38877464404185294878323159152, −2.54401642319011399557920759889, −1.43423414088871078406280124121, 0, 1.43423414088871078406280124121, 2.54401642319011399557920759889, 3.38877464404185294878323159152, 4.64577805751899849665319383092, 5.36650684226892599309211492342, 6.33738862277121156447234649392, 7.21285497786280286801180495079, 7.62887873867370361852526319092, 8.574049298258472467392444731185

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