Properties

Label 2-19800-1.1-c1-0-9
Degree $2$
Conductor $19800$
Sign $1$
Analytic cond. $158.103$
Root an. cond. $12.5739$
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  − 11-s − 2·13-s + 6·17-s + 4·23-s − 2·29-s + 10·37-s − 6·41-s + 8·43-s − 4·47-s − 7·49-s − 6·53-s + 12·59-s + 2·61-s − 4·67-s − 12·71-s + 14·73-s + 16·79-s − 12·83-s − 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.834·23-s − 0.371·29-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s − 49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.488·67-s − 1.42·71-s + 1.63·73-s + 1.80·79-s − 1.31·83-s − 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(158.103\)
Root analytic conductor: \(12.5739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 19800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.115367578\)
\(L(\frac12)\) \(\approx\) \(2.115367578\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.66825713621318, −15.04162545771013, −14.55435216552812, −14.23441915701990, −13.40356850989767, −12.87990022156605, −12.51218826900467, −11.76436278368414, −11.33632724014619, −10.68195563838813, −10.01871225229276, −9.623178397889004, −9.039826488889793, −8.187480776521842, −7.787257256361616, −7.214651836525235, −6.511082407945337, −5.802972746926757, −5.211713616676850, −4.662099592292870, −3.794959262065611, −3.106252640028647, −2.476586398044074, −1.492737076914549, −0.6243313477528390, 0.6243313477528390, 1.492737076914549, 2.476586398044074, 3.106252640028647, 3.794959262065611, 4.662099592292870, 5.211713616676850, 5.802972746926757, 6.511082407945337, 7.214651836525235, 7.787257256361616, 8.187480776521842, 9.039826488889793, 9.623178397889004, 10.01871225229276, 10.68195563838813, 11.33632724014619, 11.76436278368414, 12.51218826900467, 12.87990022156605, 13.40356850989767, 14.23441915701990, 14.55435216552812, 15.04162545771013, 15.66825713621318

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