Properties

Label 2-1248-1.1-c1-0-19
Degree $2$
Conductor $1248$
Sign $-1$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.23·5-s − 3.23·7-s + 9-s − 2·11-s − 13-s − 3.23·15-s − 4.47·17-s − 0.763·19-s + 3.23·21-s − 6.47·23-s + 5.47·25-s − 27-s − 4.47·29-s + 5.70·31-s + 2·33-s − 10.4·35-s − 8.47·37-s + 39-s − 3.23·41-s + 2.47·43-s + 3.23·45-s + 10.9·47-s + 3.47·49-s + 4.47·51-s − 0.472·53-s − 6.47·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.44·5-s − 1.22·7-s + 0.333·9-s − 0.603·11-s − 0.277·13-s − 0.835·15-s − 1.08·17-s − 0.175·19-s + 0.706·21-s − 1.34·23-s + 1.09·25-s − 0.192·27-s − 0.830·29-s + 1.02·31-s + 0.348·33-s − 1.77·35-s − 1.39·37-s + 0.160·39-s − 0.505·41-s + 0.376·43-s + 0.482·45-s + 1.59·47-s + 0.496·49-s + 0.626·51-s − 0.0648·53-s − 0.872·55-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 + 3.23T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + 0.763T + 19T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 + 3.23T + 41T^{2} \)
43 \( 1 - 2.47T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 - 0.472T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 + 8.47T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 4.47T + 97T^{2} \)
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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

−9.433145564637886026903057584221, −8.759874431948671734139567460138, −7.43425858912194942112343560581, −6.52275900808355859074240311884, −6.02272004970955032967360176030, −5.29396245278535767365914782195, −4.16066266332991655782669416607, −2.80243939547291842600376812437, −1.87962937083684183301061413624, 0, 1.87962937083684183301061413624, 2.80243939547291842600376812437, 4.16066266332991655782669416607, 5.29396245278535767365914782195, 6.02272004970955032967360176030, 6.52275900808355859074240311884, 7.43425858912194942112343560581, 8.759874431948671734139567460138, 9.433145564637886026903057584221

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