Properties

Degree $2$
Conductor $3264$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s − 4·11-s + 2·13-s + 2·15-s + 17-s + 4·19-s − 25-s + 27-s + 10·29-s − 8·31-s − 4·33-s + 2·37-s + 2·39-s + 10·41-s + 12·43-s + 2·45-s − 7·49-s + 51-s − 6·53-s − 8·55-s + 4·57-s + 12·59-s + 10·61-s + 4·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s + 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s + 0.298·45-s − 49-s + 0.140·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 1.56·59-s + 1.28·61-s + 0.496·65-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3264} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3264,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.809991036\)
\(L(\frac12)\) \(\approx\) \(2.809991036\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.658113684153939465660001552715, −7.890763986910657579747142309970, −7.34292617858622580122915984824, −6.29236664701447308952901966222, −5.62906231402718523510730620943, −4.91746541938042376263088347369, −3.84610425166553964062381179508, −2.86767294411575254972359726206, −2.21434037861733189800471556077, −1.01738437769258744176787321955, 1.01738437769258744176787321955, 2.21434037861733189800471556077, 2.86767294411575254972359726206, 3.84610425166553964062381179508, 4.91746541938042376263088347369, 5.62906231402718523510730620943, 6.29236664701447308952901966222, 7.34292617858622580122915984824, 7.890763986910657579747142309970, 8.658113684153939465660001552715

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