Properties

Label 2-2850-1.1-c1-0-24
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 − 0.449·7-s + 8-s + 9-s − 1.44·11-s + 12-s + 2.44·13-s − 0.449·14-s + 16-s − 0.449·17-s + 18-s + 19-s − 0.449·21-s − 1.44·22-s − 23-s + 24-s + 2.44·26-s + 27-s − 0.449·28-s + 10.3·29-s − 3·31-s + 32-s − 1.44·33-s − 0.449·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.169·7-s + 0.353·8-s + 0.333·9-s − 0.437·11-s + 0.288·12-s + 0.679·13-s − 0.120·14-s + 0.250·16-s − 0.109·17-s + 0.235·18-s + 0.229·19-s − 0.0980·21-s − 0.309·22-s − 0.208·23-s + 0.204·24-s + 0.480·26-s + 0.192·27-s − 0.0849·28-s + 1.92·29-s − 0.538·31-s + 0.176·32-s − 0.252·33-s − 0.0770·34-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.735453367\)
\(L(\frac12)\) \(\approx\) \(3.735453367\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.449T + 7T^{2} \)
11 \( 1 + 1.44T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + 0.449T + 17T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 - 8.89T + 41T^{2} \)
43 \( 1 - 2.44T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + 2.55T + 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 + 4.55T + 61T^{2} \)
67 \( 1 + 9.24T + 67T^{2} \)
71 \( 1 + 6.44T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 6.44T + 97T^{2} \)
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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

−8.737758727543438572287638875497, −7.911963501914263737797331977426, −7.34322251645105806821781545156, −6.32428577261191538069835441484, −5.81313881437308006709748361570, −4.68096876548028483421755905033, −4.08927976574825953133204088878, −3.06105256458612005816213438089, −2.44451831551702880233888861245, −1.11114235008707996922985062677, 1.11114235008707996922985062677, 2.44451831551702880233888861245, 3.06105256458612005816213438089, 4.08927976574825953133204088878, 4.68096876548028483421755905033, 5.81313881437308006709748361570, 6.32428577261191538069835441484, 7.34322251645105806821781545156, 7.911963501914263737797331977426, 8.737758727543438572287638875497

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