Properties

Label 2-1248-1.1-c1-0-5
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 + 2·7-s + 9-s + 4·11-s + 13-s − 6·17-s + 6·19-s − 2·21-s − 5·25-s − 27-s − 2·29-s − 6·31-s − 4·33-s + 10·37-s − 39-s + 8·41-s + 12·43-s + 12·47-s − 3·49-s + 6·51-s − 6·53-s − 6·57-s + 2·61-s + 2·63-s + 2·67-s − 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 1.45·17-s + 1.37·19-s − 0.436·21-s − 25-s − 0.192·27-s − 0.371·29-s − 1.07·31-s − 0.696·33-s + 1.64·37-s − 0.160·39-s + 1.24·41-s + 1.82·43-s + 1.75·47-s − 3/7·49-s + 0.840·51-s − 0.824·53-s − 0.794·57-s + 0.256·61-s + 0.251·63-s + 0.244·67-s − 0.949·71-s + 1.63·73-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: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586183076\)
\(L(\frac12)\) \(\approx\) \(1.586183076\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−9.342068647681432314010660944255, −9.269820194810572915186635399883, −7.920126496569155684931452044745, −7.26948453317897544910090398161, −6.26794152529161447517798404710, −5.59232004512023197121827602220, −4.48413639840570454203249914552, −3.84092207155881713644722109378, −2.22728463189050631057902889299, −1.02393888089349693967293608434, 1.02393888089349693967293608434, 2.22728463189050631057902889299, 3.84092207155881713644722109378, 4.48413639840570454203249914552, 5.59232004512023197121827602220, 6.26794152529161447517798404710, 7.26948453317897544910090398161, 7.920126496569155684931452044745, 9.269820194810572915186635399883, 9.342068647681432314010660944255

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