Properties

Label 2-2646-1.1-c1-0-17
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + 5·11-s + 16-s − 2·17-s + 19-s − 20-s + 5·22-s − 23-s − 4·25-s + 4·29-s + 9·31-s + 32-s − 2·34-s + 5·37-s + 38-s − 40-s + 9·41-s − 10·43-s + 5·44-s − 46-s − 6·47-s − 4·50-s + 12·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.50·11-s + 1/4·16-s − 0.485·17-s + 0.229·19-s − 0.223·20-s + 1.06·22-s − 0.208·23-s − 4/5·25-s + 0.742·29-s + 1.61·31-s + 0.176·32-s − 0.342·34-s + 0.821·37-s + 0.162·38-s − 0.158·40-s + 1.40·41-s − 1.52·43-s + 0.753·44-s − 0.147·46-s − 0.875·47-s − 0.565·50-s + 1.64·53-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: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.940488761\)
\(L(\frac12)\) \(\approx\) \(2.940488761\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.758419521307545992457221280358, −8.073840738987368495962393102681, −7.15629851020103523042213387688, −6.47784583599995832693494924500, −5.87450213364309522501266401702, −4.69069334189536458166730202416, −4.15202289907072344630083460584, −3.35511404251805033708538922757, −2.26551691900571842895890768283, −1.01936668222152272292011651270, 1.01936668222152272292011651270, 2.26551691900571842895890768283, 3.35511404251805033708538922757, 4.15202289907072344630083460584, 4.69069334189536458166730202416, 5.87450213364309522501266401702, 6.47784583599995832693494924500, 7.15629851020103523042213387688, 8.073840738987368495962393102681, 8.758419521307545992457221280358

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