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Properties

Label 2-96600-1.1-c1-0-2
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 + 2·13-s − 4·17-s − 2·19-s + 21-s + 23-s − 27-s − 2·31-s + 2·33-s − 10·37-s − 2·39-s + 2·41-s − 6·43-s + 2·47-s + 49-s + 4·51-s − 6·53-s + 2·57-s + 4·59-s + 14·61-s − 63-s + 10·67-s − 69-s − 2·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.970·17-s − 0.458·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.359·31-s + 0.348·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.914·43-s + 0.291·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s + 0.264·57-s + 0.520·59-s + 1.79·61-s − 0.125·63-s + 1.22·67-s − 0.120·69-s − 0.237·71-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

\(L(1)\) \(\approx\) \(0.6998824942\)
\(L(\frac12)\) \(\approx\) \(0.6998824942\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−13.66427775865443, −13.18073604482591, −12.85392400315622, −12.43937717939211, −11.70095358978971, −11.37541291925500, −10.80133939623542, −10.44114692586395, −9.955053401314331, −9.338715265773831, −8.759771123370947, −8.381099956316040, −7.766382324163626, −6.987483098196287, −6.761582600444872, −6.199878357964022, −5.546090174855689, −5.123733127376684, −4.546964838041926, −3.821895505897948, −3.425786914917777, −2.527252131768215, −2.012307229017864, −1.199069527864392, −0.2858977987716747, 0.2858977987716747, 1.199069527864392, 2.012307229017864, 2.527252131768215, 3.425786914917777, 3.821895505897948, 4.546964838041926, 5.123733127376684, 5.546090174855689, 6.199878357964022, 6.761582600444872, 6.987483098196287, 7.766382324163626, 8.381099956316040, 8.759771123370947, 9.338715265773831, 9.955053401314331, 10.44114692586395, 10.80133939623542, 11.37541291925500, 11.70095358978971, 12.43937717939211, 12.85392400315622, 13.18073604482591, 13.66427775865443

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