Properties

Label 2-56550-1.1-c1-0-39
Degree $2$
Conductor $56550$
Sign $1$
Analytic cond. $451.554$
Root an. cond. $21.2498$
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 + 6-s + 4·7-s + 8-s + 9-s − 4·11-s + 12-s − 13-s + 4·14-s + 16-s + 6·17-s + 18-s − 4·19-s + 4·21-s − 4·22-s + 8·23-s + 24-s − 26-s + 27-s + 4·28-s + 29-s + 8·31-s + 32-s − 4·33-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.872·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.755·28-s + 0.185·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13 \cdot 29\)
Sign: $1$
Analytic conductor: \(451.554\)
Root analytic conductor: \(21.2498\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 56550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.189660678\)
\(L(\frac12)\) \(\approx\) \(7.189660678\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
29 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.57326600855063, −13.76776408355041, −13.53827566381161, −12.92443386651432, −12.41158536623502, −11.93359130077254, −11.37035461994069, −10.78003337252583, −10.42803560321623, −9.920328966658889, −9.096736801600945, −8.480419024071078, −8.090566166139862, −7.504577870615798, −7.311104431100057, −6.345018568455824, −5.722184363816121, −5.124168283012073, −4.680657370256367, −4.307751357164505, −3.335233441851698, −2.782765352875505, −2.323284636731266, −1.475666216215628, −0.8302460501315557, 0.8302460501315557, 1.475666216215628, 2.323284636731266, 2.782765352875505, 3.335233441851698, 4.307751357164505, 4.680657370256367, 5.124168283012073, 5.722184363816121, 6.345018568455824, 7.311104431100057, 7.504577870615798, 8.090566166139862, 8.480419024071078, 9.096736801600945, 9.920328966658889, 10.42803560321623, 10.78003337252583, 11.37035461994069, 11.93359130077254, 12.41158536623502, 12.92443386651432, 13.53827566381161, 13.76776408355041, 14.57326600855063

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