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Properties

Label 2-96600-1.1-c1-0-30
Degree $2$
Conductor $96600$
Sign $1$
Analytic cond. $771.354$
Root an. cond. $27.7732$
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 + 7-s + 9-s − 2·11-s + 6·13-s + 6·17-s − 6·19-s + 21-s − 23-s + 27-s + 2·29-s + 8·31-s − 2·33-s − 8·37-s + 6·39-s + 6·41-s − 10·43-s + 8·47-s + 49-s + 6·51-s + 4·53-s − 6·57-s + 8·59-s + 63-s − 14·67-s − 69-s + 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 1.45·17-s − 1.37·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.348·33-s − 1.31·37-s + 0.960·39-s + 0.937·41-s − 1.52·43-s + 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.549·53-s − 0.794·57-s + 1.04·59-s + 0.125·63-s − 1.71·67-s − 0.120·69-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(771.354\)
Root analytic conductor: \(27.7732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.870405319\)
\(L(\frac12)\) \(\approx\) \(3.870405319\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.85950627460044, −13.24874046976483, −13.04517164028516, −12.25056209798030, −11.95429783106806, −11.32253194226432, −10.66902082880042, −10.35148034365622, −10.00714136319934, −9.211599607514987, −8.556667458083075, −8.380352471941746, −8.004472516876096, −7.257622693931739, −6.767367172953595, −6.066773532012684, −5.685325921349114, −5.055220479342194, −4.220824038172893, −3.999817239484936, −3.142379509926768, −2.797347411781508, −1.891762983774012, −1.374952752925586, −0.6187723190259065, 0.6187723190259065, 1.374952752925586, 1.891762983774012, 2.797347411781508, 3.142379509926768, 3.999817239484936, 4.220824038172893, 5.055220479342194, 5.685325921349114, 6.066773532012684, 6.767367172953595, 7.257622693931739, 8.004472516876096, 8.380352471941746, 8.556667458083075, 9.211599607514987, 10.00714136319934, 10.35148034365622, 10.66902082880042, 11.32253194226432, 11.95429783106806, 12.25056209798030, 13.04517164028516, 13.24874046976483, 13.85950627460044

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