Properties

Label 2-33800-1.1-c1-0-4
Degree $2$
Conductor $33800$
Sign $1$
Analytic cond. $269.894$
Root an. cond. $16.4284$
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 + 3·7-s − 2·9-s + 11-s − 17-s − 19-s + 3·21-s + 5·23-s − 5·27-s + 3·29-s + 8·31-s + 33-s − 9·37-s − 5·41-s − 9·43-s − 8·47-s + 2·49-s − 51-s + 10·53-s − 57-s + 3·59-s − 5·61-s − 6·63-s − 3·67-s + 5·69-s − 71-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.301·11-s − 0.242·17-s − 0.229·19-s + 0.654·21-s + 1.04·23-s − 0.962·27-s + 0.557·29-s + 1.43·31-s + 0.174·33-s − 1.47·37-s − 0.780·41-s − 1.37·43-s − 1.16·47-s + 2/7·49-s − 0.140·51-s + 1.37·53-s − 0.132·57-s + 0.390·59-s − 0.640·61-s − 0.755·63-s − 0.366·67-s + 0.601·69-s − 0.118·71-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33800\)    =    \(2^{3} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(269.894\)
Root analytic conductor: \(16.4284\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.099749552\)
\(L(\frac12)\) \(\approx\) \(3.099749552\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.02514918352432, −14.50049103199861, −13.87967139863717, −13.64878064256248, −13.04328191326492, −12.20449023635154, −11.75656155411521, −11.39937069167863, −10.74773807640253, −10.23868507294990, −9.571439249744407, −8.871553054249638, −8.410568376558006, −8.233772783803737, −7.447998780905265, −6.747995627351736, −6.318539790245044, −5.322378741168100, −4.995184973399720, −4.373832391844162, −3.476241573112380, −3.026747465141129, −2.154830923451995, −1.614999657732528, −0.6375324684474031, 0.6375324684474031, 1.614999657732528, 2.154830923451995, 3.026747465141129, 3.476241573112380, 4.373832391844162, 4.995184973399720, 5.322378741168100, 6.318539790245044, 6.747995627351736, 7.447998780905265, 8.233772783803737, 8.410568376558006, 8.871553054249638, 9.571439249744407, 10.23868507294990, 10.74773807640253, 11.39937069167863, 11.75656155411521, 12.20449023635154, 13.04328191326492, 13.64878064256248, 13.87967139863717, 14.50049103199861, 15.02514918352432

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