Properties

Label 2-2850-1.1-c1-0-53
Degree $2$
Conductor $2850$
Sign $-1$
Analytic cond. $22.7573$
Root an. cond. $4.77046$
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 + 6-s − 4·7-s + 8-s + 9-s − 11-s + 12-s − 4·14-s + 16-s − 8·17-s + 18-s + 19-s − 4·21-s − 22-s − 3·23-s + 24-s + 27-s − 4·28-s − 29-s + 31-s + 32-s − 33-s − 8·34-s + 36-s − 2·37-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 − 0.301·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 0.229·19-s − 0.872·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s − 0.185·29-s + 0.179·31-s + 0.176·32-s − 0.174·33-s − 1.37·34-s + 1/6·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(22.7573\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−8.531417878744045818807268828279, −7.45654365124277994811016836720, −6.70756534175442960648465476997, −6.30080590719207160691412332497, −5.24306934453883269904147412330, −4.30801399964828431306303840795, −3.53698332852679765889221274218, −2.80683694786922496217827209499, −1.93092679700969988135790454085, 0, 1.93092679700969988135790454085, 2.80683694786922496217827209499, 3.53698332852679765889221274218, 4.30801399964828431306303840795, 5.24306934453883269904147412330, 6.30080590719207160691412332497, 6.70756534175442960648465476997, 7.45654365124277994811016836720, 8.531417878744045818807268828279

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