Properties

Label 2-301530-1.1-c1-0-63
Degree $2$
Conductor $301530$
Sign $-1$
Analytic cond. $2407.72$
Root an. cond. $49.0686$
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 − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 4·11-s − 12-s + 2·13-s − 15-s + 16-s − 2·17-s + 18-s + 19-s + 20-s − 4·22-s − 24-s + 25-s + 2·26-s − 27-s + 6·29-s − 30-s + 4·31-s + 32-s + 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.852·22-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(301530\)    =    \(2 \cdot 3 \cdot 5 \cdot 19 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(2407.72\)
Root analytic conductor: \(49.0686\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{301530} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 301530,\ (\ :1/2),\ -1)\)

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

−12.99649214683175, −12.56080330910013, −12.11199019953257, −11.41849599370906, −11.13751051344301, −10.82119474340994, −10.19337339188005, −9.900820060199529, −9.353242871135786, −8.735372933635644, −8.154899786706738, −7.698624204553013, −7.364305109768503, −6.461903822270885, −6.247060711365598, −6.021966416672445, −5.156033168711019, −4.913377476776621, −4.509148715994160, −3.859765875949874, −3.124562861104052, −2.730094424781593, −2.195977677148509, −1.441707829064450, −0.8675080387887004, 0, 0.8675080387887004, 1.441707829064450, 2.195977677148509, 2.730094424781593, 3.124562861104052, 3.859765875949874, 4.509148715994160, 4.913377476776621, 5.156033168711019, 6.021966416672445, 6.247060711365598, 6.461903822270885, 7.364305109768503, 7.698624204553013, 8.154899786706738, 8.735372933635644, 9.353242871135786, 9.900820060199529, 10.19337339188005, 10.82119474340994, 11.13751051344301, 11.41849599370906, 12.11199019953257, 12.56080330910013, 12.99649214683175

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