Properties

Label 2-56350-1.1-c1-0-30
Degree $2$
Conductor $56350$
Sign $-1$
Analytic cond. $449.957$
Root an. cond. $21.2121$
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 − 8-s − 3·9-s + 4·11-s − 3·13-s + 16-s − 17-s + 3·18-s − 4·22-s + 23-s + 3·26-s + 4·31-s − 32-s + 34-s − 3·36-s − 11·37-s + 10·41-s + 2·43-s + 4·44-s − 46-s + 11·47-s − 3·52-s − 53-s + 8·59-s + 8·61-s − 4·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s − 0.832·13-s + 1/4·16-s − 0.242·17-s + 0.707·18-s − 0.852·22-s + 0.208·23-s + 0.588·26-s + 0.718·31-s − 0.176·32-s + 0.171·34-s − 1/2·36-s − 1.80·37-s + 1.56·41-s + 0.304·43-s + 0.603·44-s − 0.147·46-s + 1.60·47-s − 0.416·52-s − 0.137·53-s + 1.04·59-s + 1.02·61-s − 0.508·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56350\)    =    \(2 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(449.957\)
Root analytic conductor: \(21.2121\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 56350,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.66486563306522, −14.14613773049607, −13.87870496556819, −13.04094349652634, −12.38747465359559, −12.00347565812698, −11.58258052107753, −11.05551310619002, −10.53892659882311, −9.932712500925060, −9.417286791080220, −8.887450104432504, −8.589865696513544, −7.965956899014067, −7.182123879952363, −6.954589001779276, −6.234051316674914, −5.671071625998016, −5.162955997763484, −4.244863261664662, −3.812949143171800, −2.829116064593464, −2.528833032049554, −1.624939711839506, −0.8653034275834763, 0, 0.8653034275834763, 1.624939711839506, 2.528833032049554, 2.829116064593464, 3.812949143171800, 4.244863261664662, 5.162955997763484, 5.671071625998016, 6.234051316674914, 6.954589001779276, 7.182123879952363, 7.965956899014067, 8.589865696513544, 8.887450104432504, 9.417286791080220, 9.932712500925060, 10.53892659882311, 11.05551310619002, 11.58258052107753, 12.00347565812698, 12.38747465359559, 13.04094349652634, 13.87870496556819, 14.14613773049607, 14.66486563306522

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