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Properties

Label 2-96600-1.1-c1-0-37
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 + 6·17-s − 4·19-s + 21-s − 23-s + 27-s + 2·29-s + 10·31-s − 2·33-s + 8·37-s + 6·41-s + 12·43-s − 4·47-s + 49-s + 6·51-s + 8·53-s − 4·57-s − 10·59-s + 6·61-s + 63-s + 8·67-s − 69-s + 4·71-s + 16·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.45·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s + 0.371·29-s + 1.79·31-s − 0.348·33-s + 1.31·37-s + 0.937·41-s + 1.82·43-s − 0.583·47-s + 1/7·49-s + 0.840·51-s + 1.09·53-s − 0.529·57-s − 1.30·59-s + 0.768·61-s + 0.125·63-s + 0.977·67-s − 0.120·69-s + 0.474·71-s + 1.87·73-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

\(L(1)\) \(\approx\) \(4.273103100\)
\(L(\frac12)\) \(\approx\) \(4.273103100\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.86349044132499, −13.39397471561635, −12.75420256910420, −12.36415422757373, −12.01848274876420, −11.21009122708698, −10.83916837075703, −10.31258743380989, −9.738730271347814, −9.472068428171393, −8.677803430327586, −8.224967681678209, −7.743601417957070, −7.588847598922776, −6.580817241884751, −6.266722732924697, −5.542773658568845, −5.030189706237801, −4.362605258378427, −3.938883711837191, −3.202553099805077, −2.483344800417646, −2.271487346064031, −1.138855660084594, −0.7185479549938094, 0.7185479549938094, 1.138855660084594, 2.271487346064031, 2.483344800417646, 3.202553099805077, 3.938883711837191, 4.362605258378427, 5.030189706237801, 5.542773658568845, 6.266722732924697, 6.580817241884751, 7.588847598922776, 7.743601417957070, 8.224967681678209, 8.677803430327586, 9.472068428171393, 9.738730271347814, 10.31258743380989, 10.83916837075703, 11.21009122708698, 12.01848274876420, 12.36415422757373, 12.75420256910420, 13.39397471561635, 13.86349044132499

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