Properties

Label 2-2496-1.1-c1-0-40
Degree $2$
Conductor $2496$
Sign $-1$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
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 + 4·7-s + 9-s − 2·11-s + 13-s − 6·17-s − 4·19-s − 4·21-s − 4·23-s − 5·25-s − 27-s − 10·29-s + 8·31-s + 2·33-s + 2·37-s − 39-s − 4·43-s − 2·47-s + 9·49-s + 6·51-s + 2·53-s + 4·57-s + 10·59-s − 10·61-s + 4·63-s + 8·67-s + 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 1.45·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s − 25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.160·39-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.840·51-s + 0.274·53-s + 0.529·57-s + 1.30·59-s − 1.28·61-s + 0.503·63-s + 0.977·67-s + 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.324902971952138704605785202150, −7.939260681586206397190079639535, −7.00034310960812354176480720233, −6.11140808730305310116200445706, −5.39682733407529528593800657224, −4.53465105360237919334130772304, −4.01593017646071294816935896556, −2.36195647663830288126441129549, −1.64247750087938000791019969761, 0, 1.64247750087938000791019969761, 2.36195647663830288126441129549, 4.01593017646071294816935896556, 4.53465105360237919334130772304, 5.39682733407529528593800657224, 6.11140808730305310116200445706, 7.00034310960812354176480720233, 7.939260681586206397190079639535, 8.324902971952138704605785202150

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