Properties

Label 2-75810-1.1-c1-0-6
Degree $2$
Conductor $75810$
Sign $1$
Analytic cond. $605.345$
Root an. cond. $24.6037$
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  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 4·11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s − 6·17-s + 18-s − 20-s − 21-s − 4·22-s − 8·23-s + 24-s + 25-s + 2·26-s + 27-s − 28-s − 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75810\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(605.345\)
Root analytic conductor: \(24.6037\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75810} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75810,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.993122809\)
\(L(\frac12)\) \(\approx\) \(1.993122809\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
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

−13.89293907144860, −13.66819180958431, −13.05040841724218, −12.69250620638907, −12.28645149775864, −11.51518182977567, −11.11317236945385, −10.64395627982204, −10.12097868354643, −9.490871902474075, −8.968535247073823, −8.327045761538707, −7.858986202764069, −7.478047990542749, −6.828045511808433, −6.120619645879228, −5.873541269657000, −5.001102640777862, −4.451396178053970, −3.949995859931644, −3.461110801203180, −2.670905749687500, −2.302642090542788, −1.576730723225287, −0.3677304605337877, 0.3677304605337877, 1.576730723225287, 2.302642090542788, 2.670905749687500, 3.461110801203180, 3.949995859931644, 4.451396178053970, 5.001102640777862, 5.873541269657000, 6.120619645879228, 6.828045511808433, 7.478047990542749, 7.858986202764069, 8.327045761538707, 8.968535247073823, 9.490871902474075, 10.12097868354643, 10.64395627982204, 11.11317236945385, 11.51518182977567, 12.28645149775864, 12.69250620638907, 13.05040841724218, 13.66819180958431, 13.89293907144860

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