Properties

Label 2-2850-1.1-c1-0-1
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 − 3.35·7-s − 8-s + 9-s − 1.61·11-s − 12-s + 1.35·13-s + 3.35·14-s + 16-s − 6.96·17-s − 18-s − 19-s + 3.35·21-s + 1.61·22-s + 1.35·23-s + 24-s − 1.35·26-s − 27-s − 3.35·28-s + 3.61·29-s − 2.31·31-s − 32-s + 1.61·33-s + 6.96·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.26·7-s − 0.353·8-s + 0.333·9-s − 0.486·11-s − 0.288·12-s + 0.374·13-s + 0.895·14-s + 0.250·16-s − 1.68·17-s − 0.235·18-s − 0.229·19-s + 0.731·21-s + 0.343·22-s + 0.281·23-s + 0.204·24-s − 0.264·26-s − 0.192·27-s − 0.633·28-s + 0.670·29-s − 0.415·31-s − 0.176·32-s + 0.280·33-s + 1.19·34-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

\(L(1)\) \(\approx\) \(0.4959807754\)
\(L(\frac12)\) \(\approx\) \(0.4959807754\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.35T + 7T^{2} \)
11 \( 1 + 1.61T + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 + 6.96T + 17T^{2} \)
23 \( 1 - 1.35T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 + 2.31T + 31T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + 3.35T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 4.57T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 - 1.03T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 9.92T + 67T^{2} \)
71 \( 1 + 0.775T + 71T^{2} \)
73 \( 1 - 3.22T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 2.57T + 89T^{2} \)
97 \( 1 - 1.16T + 97T^{2} \)
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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

−8.934925939383128903158202441565, −8.129583550664170780109766942340, −7.04780761997897086621194786906, −6.65662885175712153736067539703, −5.96684596844018106182777058733, −5.01914658157131205870147332853, −3.96363138790030970759103552138, −2.98295707879308733740525477006, −1.97550494486033083755389630721, −0.47332244389407452799266575899, 0.47332244389407452799266575899, 1.97550494486033083755389630721, 2.98295707879308733740525477006, 3.96363138790030970759103552138, 5.01914658157131205870147332853, 5.96684596844018106182777058733, 6.65662885175712153736067539703, 7.04780761997897086621194786906, 8.129583550664170780109766942340, 8.934925939383128903158202441565

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