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Properties

Label 2-46200-1.1-c1-0-76
Degree $2$
Conductor $46200$
Sign $-1$
Analytic cond. $368.908$
Root an. cond. $19.2070$
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 + 11-s + 2·13-s − 2·17-s + 4·19-s − 21-s + 27-s − 2·29-s − 8·31-s + 33-s + 10·37-s + 2·39-s − 6·41-s + 4·43-s + 49-s − 2·51-s + 2·53-s + 4·57-s − 12·59-s + 14·61-s − 63-s − 12·67-s + 6·73-s − 77-s − 16·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.174·33-s + 1.64·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s + 0.274·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s − 0.125·63-s − 1.46·67-s + 0.702·73-s − 0.113·77-s − 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(368.908\)
Root analytic conductor: \(19.2070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.81950389027073, −14.40841725822562, −13.76967684878021, −13.37306110963089, −12.93936215300049, −12.38382127443138, −11.78076439130387, −11.14936379661498, −10.86052660802259, −9.986453764634996, −9.641902182655748, −9.040948578235698, −8.719571598878805, −7.942569842465655, −7.495329499040097, −6.912499616482170, −6.348167584235292, −5.683870719465269, −5.171860579584662, −4.203199113849742, −3.921535389438634, −3.128889524634798, −2.638751293461652, −1.746080927866114, −1.112373619261541, 0, 1.112373619261541, 1.746080927866114, 2.638751293461652, 3.128889524634798, 3.921535389438634, 4.203199113849742, 5.171860579584662, 5.683870719465269, 6.348167584235292, 6.912499616482170, 7.495329499040097, 7.942569842465655, 8.719571598878805, 9.040948578235698, 9.641902182655748, 9.986453764634996, 10.86052660802259, 11.14936379661498, 11.78076439130387, 12.38382127443138, 12.93936215300049, 13.37306110963089, 13.76967684878021, 14.40841725822562, 14.81950389027073

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