Properties

Label 2-75690-1.1-c1-0-21
Degree $2$
Conductor $75690$
Sign $-1$
Analytic cond. $604.387$
Root an. cond. $24.5842$
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  − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 2·13-s + 4·14-s + 16-s + 6·17-s + 4·19-s + 20-s + 25-s − 2·26-s − 4·28-s − 8·31-s − 32-s − 6·34-s − 4·35-s − 2·37-s − 4·38-s − 40-s − 6·41-s + 4·43-s + 9·49-s − 50-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.755·28-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.141·50-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75690\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(604.387\)
Root analytic conductor: \(24.5842\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75690,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
29 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.45023179515949, −13.75318066821554, −13.09544593432898, −12.99463253805780, −12.18443634551117, −11.90680483860723, −11.25254135991530, −10.56209408735179, −10.12785820438042, −9.856484649877465, −9.227120311644552, −8.951338084473071, −8.278051038260176, −7.542051736160220, −7.211707083198986, −6.608706321036608, −6.067334030075017, −5.554423734487991, −5.185206258518577, −3.982015927896784, −3.475927089400618, −3.078307866014637, −2.363774445029134, −1.495468383169606, −0.8870903136222875, 0, 0.8870903136222875, 1.495468383169606, 2.363774445029134, 3.078307866014637, 3.475927089400618, 3.982015927896784, 5.185206258518577, 5.554423734487991, 6.067334030075017, 6.608706321036608, 7.211707083198986, 7.542051736160220, 8.278051038260176, 8.951338084473071, 9.227120311644552, 9.856484649877465, 10.12785820438042, 10.56209408735179, 11.25254135991530, 11.90680483860723, 12.18443634551117, 12.99463253805780, 13.09544593432898, 13.75318066821554, 14.45023179515949

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