Properties

Label 2-4950-1.1-c1-0-74
Degree $2$
Conductor $4950$
Sign $-1$
Analytic cond. $39.5259$
Root an. cond. $6.28696$
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  + 2-s + 4-s + 8-s − 11-s − 6·13-s + 16-s + 2·17-s − 4·19-s − 22-s − 6·26-s + 10·29-s + 32-s + 2·34-s − 6·37-s − 4·38-s − 2·41-s − 4·43-s − 44-s − 8·47-s − 7·49-s − 6·52-s − 10·53-s + 10·58-s + 4·59-s − 2·61-s + 64-s + 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.213·22-s − 1.17·26-s + 1.85·29-s + 0.176·32-s + 0.342·34-s − 0.986·37-s − 0.648·38-s − 0.312·41-s − 0.609·43-s − 0.150·44-s − 1.16·47-s − 49-s − 0.832·52-s − 1.37·53-s + 1.31·58-s + 0.520·59-s − 0.256·61-s + 1/8·64-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4950\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(39.5259\)
Root analytic conductor: \(6.28696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4950,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−7.953519059448833265921148977490, −6.87516314576116310072723973234, −6.61930482181016241055363584086, −5.50519254084154730512895831306, −4.91176036475039380235487105270, −4.34547747847136462528881406880, −3.22266685034392941829856826252, −2.59758594639330277923535938645, −1.61525671840520505272701261819, 0, 1.61525671840520505272701261819, 2.59758594639330277923535938645, 3.22266685034392941829856826252, 4.34547747847136462528881406880, 4.91176036475039380235487105270, 5.50519254084154730512895831306, 6.61930482181016241055363584086, 6.87516314576116310072723973234, 7.953519059448833265921148977490

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