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Properties

Label 2-96600-1.1-c1-0-27
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 + 6·17-s + 4·19-s − 21-s − 23-s + 27-s + 10·29-s − 8·31-s + 2·37-s − 2·39-s − 6·41-s + 8·47-s + 49-s + 6·51-s + 10·53-s + 4·57-s + 12·59-s − 2·61-s − 63-s − 8·67-s − 69-s − 12·71-s + 6·73-s − 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 1.16·47-s + 1/7·49-s + 0.840·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.125·63-s − 0.977·67-s − 0.120·69-s − 1.42·71-s + 0.702·73-s − 0.900·79-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

\(L(1)\) \(\approx\) \(3.368024391\)
\(L(\frac12)\) \(\approx\) \(3.368024391\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.80289487560884, −13.40564275332467, −12.80085310967840, −12.31881715180020, −11.87159049179397, −11.58128369759036, −10.51501144032034, −10.36690241159137, −9.774242638775424, −9.450395531112335, −8.690740924113540, −8.431974913322687, −7.651977669658223, −7.300640251265560, −6.916252700642150, −6.087680912886023, −5.553167652933265, −5.132870366857527, −4.352740681013410, −3.814999007537533, −3.126866452762963, −2.828427074136689, −2.033193274670080, −1.250735030282049, −0.5954605327486724, 0.5954605327486724, 1.250735030282049, 2.033193274670080, 2.828427074136689, 3.126866452762963, 3.814999007537533, 4.352740681013410, 5.132870366857527, 5.553167652933265, 6.087680912886023, 6.916252700642150, 7.300640251265560, 7.651977669658223, 8.431974913322687, 8.690740924113540, 9.450395531112335, 9.774242638775424, 10.36690241159137, 10.51501144032034, 11.58128369759036, 11.87159049179397, 12.31881715180020, 12.80085310967840, 13.40564275332467, 13.80289487560884

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