Properties

Label 2-1248-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 2·7-s + 9-s − 2·11-s − 13-s + 2·15-s + 6·17-s + 2·19-s + 2·21-s − 25-s + 27-s + 2·29-s − 2·31-s − 2·33-s + 4·35-s + 10·37-s − 39-s + 2·41-s − 8·43-s + 2·45-s + 2·47-s − 3·49-s + 6·51-s − 2·53-s − 4·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.516·15-s + 1.45·17-s + 0.458·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.676·35-s + 1.64·37-s − 0.160·39-s + 0.312·41-s − 1.21·43-s + 0.298·45-s + 0.291·47-s − 3/7·49-s + 0.840·51-s − 0.274·53-s − 0.539·55-s + 0.264·57-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: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.601846172\)
\(L(\frac12)\) \(\approx\) \(2.601846172\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−9.787824641257285407093264142809, −8.949314783945752446588422997654, −7.896842730015818559963999252704, −7.59875356765337238034292890321, −6.28159176985699903858961238086, −5.44020310407890724415031834900, −4.68403548258509043319774249783, −3.37296560686463484601752098732, −2.38879148145950189793255761663, −1.33803744783185283027867448010, 1.33803744783185283027867448010, 2.38879148145950189793255761663, 3.37296560686463484601752098732, 4.68403548258509043319774249783, 5.44020310407890724415031834900, 6.28159176985699903858961238086, 7.59875356765337238034292890321, 7.896842730015818559963999252704, 8.949314783945752446588422997654, 9.787824641257285407093264142809

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