Properties

Label 2-3264-1.1-c1-0-47
Degree $2$
Conductor $3264$
Sign $-1$
Analytic cond. $26.0631$
Root an. cond. $5.10521$
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  − 3-s + 2·7-s + 9-s − 2·13-s − 17-s + 4·19-s − 2·21-s − 6·23-s − 5·25-s − 27-s − 10·31-s − 8·37-s + 2·39-s + 6·41-s + 4·43-s + 12·47-s − 3·49-s + 51-s − 6·53-s − 4·57-s + 12·59-s − 8·61-s + 2·63-s + 4·67-s + 6·69-s + 6·71-s + 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.242·17-s + 0.917·19-s − 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.79·31-s − 1.31·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s + 0.712·71-s + 0.234·73-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: $-1$
Analytic conductor: \(26.0631\)
Root analytic conductor: \(5.10521\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3264,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
17 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−8.115102227964514339079390978935, −7.51476003917695097413986406817, −6.88843577578684808918223131504, −5.70964004788136228366901344496, −5.44237790561600373941790334940, −4.40279032004747662521271149929, −3.72111015541234406836981026227, −2.37227056817861814340285837209, −1.47870270507371948822204128989, 0, 1.47870270507371948822204128989, 2.37227056817861814340285837209, 3.72111015541234406836981026227, 4.40279032004747662521271149929, 5.44237790561600373941790334940, 5.70964004788136228366901344496, 6.88843577578684808918223131504, 7.51476003917695097413986406817, 8.115102227964514339079390978935

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