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Properties

Label 2-96600-1.1-c1-0-17
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 − 2·13-s − 2·17-s + 21-s − 23-s + 27-s − 10·29-s − 4·31-s − 6·37-s − 2·39-s + 10·41-s + 12·43-s + 49-s − 2·51-s + 6·53-s + 4·59-s + 14·61-s + 63-s + 12·67-s − 69-s − 12·71-s − 14·73-s + 81-s − 12·83-s − 10·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.485·17-s + 0.218·21-s − 0.208·23-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.986·37-s − 0.320·39-s + 1.56·41-s + 1.82·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.520·59-s + 1.79·61-s + 0.125·63-s + 1.46·67-s − 0.120·69-s − 1.42·71-s − 1.63·73-s + 1/9·81-s − 1.31·83-s − 1.07·87-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

\(L(1)\) \(\approx\) \(2.497680636\)
\(L(\frac12)\) \(\approx\) \(2.497680636\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−13.93235839230698, −13.18527549950680, −12.83924156859590, −12.56470334442653, −11.66328845910049, −11.43353657999524, −10.84388107035543, −10.29568493901946, −9.817803939893576, −9.160191582392755, −8.914688615183044, −8.371826392524553, −7.573356721751324, −7.405301903569048, −6.929749844566946, −6.046922792845465, −5.580876252132175, −5.093920873729082, −4.199855050219327, −4.032569282163907, −3.289666664566991, −2.431242382430304, −2.170362428812065, −1.383864156583768, −0.4727227814004322, 0.4727227814004322, 1.383864156583768, 2.170362428812065, 2.431242382430304, 3.289666664566991, 4.032569282163907, 4.199855050219327, 5.093920873729082, 5.580876252132175, 6.046922792845465, 6.929749844566946, 7.405301903569048, 7.573356721751324, 8.371826392524553, 8.914688615183044, 9.160191582392755, 9.817803939893576, 10.29568493901946, 10.84388107035543, 11.43353657999524, 11.66328845910049, 12.56470334442653, 12.83924156859590, 13.18527549950680, 13.93235839230698

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