Properties

Label 2-1248-1.1-c1-0-17
Degree $2$
Conductor $1248$
Sign $-1$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
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 + 13-s + 2·17-s + 2·19-s + 2·21-s − 8·23-s − 5·25-s − 27-s + 6·29-s − 2·31-s − 6·37-s − 39-s − 4·43-s − 8·47-s − 3·49-s − 2·51-s − 6·53-s − 2·57-s − 4·59-s + 2·61-s − 2·63-s − 2·67-s + 8·69-s − 4·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.485·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s − 25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.986·37-s − 0.160·39-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.264·57-s − 0.520·59-s + 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.963·69-s − 0.474·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1248} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−9.586196101860007060944293941872, −8.404773792929967833538138718608, −7.65338740605061265659475162578, −6.61277232194412223963156068467, −6.03523704391687496414629389438, −5.14953177527932686418168257619, −4.03288470481684680307144406184, −3.13736585970066578227810367428, −1.65921780267175436554426250858, 0, 1.65921780267175436554426250858, 3.13736585970066578227810367428, 4.03288470481684680307144406184, 5.14953177527932686418168257619, 6.03523704391687496414629389438, 6.61277232194412223963156068467, 7.65338740605061265659475162578, 8.404773792929967833538138718608, 9.586196101860007060944293941872

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