Properties

Label 2-1950-1.1-c1-0-17
Degree $2$
Conductor $1950$
Sign $1$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 + 2·11-s − 12-s + 13-s + 4·14-s + 16-s + 4·17-s + 18-s + 2·19-s − 4·21-s + 2·22-s − 6·23-s − 24-s + 26-s − 27-s + 4·28-s − 2·29-s − 4·31-s + 32-s − 2·33-s + 4·34-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 + 0.603·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.426·22-s − 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.348·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.949359271\)
\(L(\frac12)\) \(\approx\) \(2.949359271\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.182648124608353586531524291120, −8.182390005868430302670141391865, −7.56623418983499887240476159616, −6.74115278039220983750967218841, −5.63860941055669312891125420954, −5.33749202386514005593291265832, −4.28561933094907687769586580986, −3.63655378791011270375144157582, −2.10777338399542844214176104623, −1.20182002010796803067299954323, 1.20182002010796803067299954323, 2.10777338399542844214176104623, 3.63655378791011270375144157582, 4.28561933094907687769586580986, 5.33749202386514005593291265832, 5.63860941055669312891125420954, 6.74115278039220983750967218841, 7.56623418983499887240476159616, 8.182390005868430302670141391865, 9.182648124608353586531524291120

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