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Properties

Label 2-96600-1.1-c1-0-35
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 4·11-s − 4·13-s − 6·17-s − 6·19-s − 21-s + 23-s + 27-s − 8·31-s − 4·33-s + 6·37-s − 4·39-s + 12·41-s + 8·43-s + 49-s − 6·51-s − 6·53-s − 6·57-s + 6·59-s − 6·61-s − 63-s + 8·67-s + 69-s + 6·71-s + 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 1.37·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.640·39-s + 1.87·41-s + 1.21·43-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.794·57-s + 0.781·59-s − 0.768·61-s − 0.125·63-s + 0.977·67-s + 0.120·69-s + 0.712·71-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: \(1\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−14.18327172828146, −13.41241486088617, −12.93864598466020, −12.70104213436169, −12.45337043039540, −11.33819819003700, −11.12420967645978, −10.52304619282123, −10.17245714125886, −9.298534059841634, −9.248639630642932, −8.633550650576966, −7.854615344220618, −7.657533289900411, −7.065123449070666, −6.434309884189236, −5.978733287900746, −5.210584078641789, −4.681103737477511, −4.206597194152251, −3.591747461271057, −2.719727044077299, −2.323762588409039, −2.040680204171349, −0.7090755845119690, 0, 0.7090755845119690, 2.040680204171349, 2.323762588409039, 2.719727044077299, 3.591747461271057, 4.206597194152251, 4.681103737477511, 5.210584078641789, 5.978733287900746, 6.434309884189236, 7.065123449070666, 7.657533289900411, 7.854615344220618, 8.633550650576966, 9.248639630642932, 9.298534059841634, 10.17245714125886, 10.52304619282123, 11.12420967645978, 11.33819819003700, 12.45337043039540, 12.70104213436169, 12.93864598466020, 13.41241486088617, 14.18327172828146

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