Properties

Label 2-1248-1.1-c1-0-20
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

−9.231246651526005478557570132192, −8.342770632372487309297815864648, −7.72597071702691609645308173983, −7.13855453193683225508653499024, −5.90056466305320169868428843649, −4.90251153820633880270104247107, −3.93592460575099925248016367060, −3.10657645810316901845317440603, −1.93631023376644655386209086893, 0, 1.93631023376644655386209086893, 3.10657645810316901845317440603, 3.93592460575099925248016367060, 4.90251153820633880270104247107, 5.90056466305320169868428843649, 7.13855453193683225508653499024, 7.72597071702691609645308173983, 8.342770632372487309297815864648, 9.231246651526005478557570132192

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