Properties

Label 2-1248-1.1-c1-0-21
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 + 2·7-s + 9-s − 6·11-s − 13-s − 2·15-s − 2·17-s − 6·19-s + 2·21-s − 25-s + 27-s − 6·29-s + 6·31-s − 6·33-s − 4·35-s + 2·37-s − 39-s − 10·41-s + 8·43-s − 2·45-s + 6·47-s − 3·49-s − 2·51-s + 6·53-s + 12·55-s − 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.516·15-s − 0.485·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.07·31-s − 1.04·33-s − 0.676·35-s + 0.328·37-s − 0.160·39-s − 1.56·41-s + 1.21·43-s − 0.298·45-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 1.61·55-s − 0.794·57-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−9.092559319520639094989226590241, −8.267121982133789942052976540936, −7.84278157109124886213393953192, −7.15914697119095734461341250765, −5.86780953840629681522673799950, −4.78263033612980466766564281150, −4.17228712390106732432301137169, −2.92133996435419900993581940697, −2.01074202882877077393828448729, 0, 2.01074202882877077393828448729, 2.92133996435419900993581940697, 4.17228712390106732432301137169, 4.78263033612980466766564281150, 5.86780953840629681522673799950, 7.15914697119095734461341250765, 7.84278157109124886213393953192, 8.267121982133789942052976540936, 9.092559319520639094989226590241

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