Properties

Label 2-2850-1.1-c1-0-18
Degree $2$
Conductor $2850$
Sign $1$
Analytic cond. $22.7573$
Root an. cond. $4.77046$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 1.07·7-s + 8-s + 9-s + 6.34·11-s − 12-s − 3.41·13-s + 1.07·14-s + 16-s + 5.41·17-s + 18-s + 19-s − 1.07·21-s + 6.34·22-s − 6.34·23-s − 24-s − 3.41·26-s − 27-s + 1.07·28-s − 0.340·29-s + 1.07·31-s + 32-s − 6.34·33-s + 5.41·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.407·7-s + 0.353·8-s + 0.333·9-s + 1.91·11-s − 0.288·12-s − 0.948·13-s + 0.288·14-s + 0.250·16-s + 1.31·17-s + 0.235·18-s + 0.229·19-s − 0.235·21-s + 1.35·22-s − 1.32·23-s − 0.204·24-s − 0.670·26-s − 0.192·27-s + 0.203·28-s − 0.0631·29-s + 0.193·31-s + 0.176·32-s − 1.10·33-s + 0.929·34-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: no
Arithmetic: yes
Character: $\chi_{2850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.864863645\)
\(L(\frac12)\) \(\approx\) \(2.864863645\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 - 6.34T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 - 5.41T + 17T^{2} \)
23 \( 1 + 6.34T + 23T^{2} \)
29 \( 1 + 0.340T + 29T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 - 7.60T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 6.34T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 0.738T + 59T^{2} \)
61 \( 1 + 2.68T + 61T^{2} \)
67 \( 1 + 2.83T + 67T^{2} \)
71 \( 1 + 2.83T + 71T^{2} \)
73 \( 1 + 6.83T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 + 0.894T + 83T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−8.855125319018892942729613298227, −7.66336827210750210261875024648, −7.28941650460627868143686261376, −6.17861540134138408290630075127, −5.86342731974160431804891755937, −4.79888333972843455806985589619, −4.17544800644602202315273018135, −3.35801230543915315290564186352, −2.04476563802979756023364690474, −1.03984464621972180411597826029, 1.03984464621972180411597826029, 2.04476563802979756023364690474, 3.35801230543915315290564186352, 4.17544800644602202315273018135, 4.79888333972843455806985589619, 5.86342731974160431804891755937, 6.17861540134138408290630075127, 7.28941650460627868143686261376, 7.66336827210750210261875024648, 8.855125319018892942729613298227

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