Properties

Label 2-1248-1.1-c1-0-15
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 − 2·5-s + 9-s + 4·11-s + 13-s + 2·15-s − 6·17-s − 8·19-s + 8·23-s − 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 10·37-s − 39-s + 6·41-s − 4·43-s − 2·45-s − 7·49-s + 6·51-s − 14·53-s − 8·55-s + 8·57-s − 12·59-s − 10·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 49-s + 0.840·51-s − 1.92·53-s − 1.07·55-s + 1.05·57-s − 1.56·59-s − 1.28·61-s − 0.248·65-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.034750489934040733997071252152, −8.672430104572291160531976657172, −7.58509571403619500018744888863, −6.53450715872245064722726153899, −6.34363053984341960282097416121, −4.66542007417636767344834083474, −4.35038208806792458849519254326, −3.16156249407593119167913883340, −1.59378843935215793575881490850, 0, 1.59378843935215793575881490850, 3.16156249407593119167913883340, 4.35038208806792458849519254326, 4.66542007417636767344834083474, 6.34363053984341960282097416121, 6.53450715872245064722726153899, 7.58509571403619500018744888863, 8.672430104572291160531976657172, 9.034750489934040733997071252152

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