Properties

Label 2-2850-1.1-c1-0-21
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 − 0.962·11-s − 12-s + 1.61·13-s + 3.35·14-s + 16-s + 0.387·17-s + 18-s + 19-s − 3.35·21-s − 0.962·22-s + 0.962·23-s − 24-s + 1.61·26-s − 27-s + 3.35·28-s + 6.96·29-s + 3.35·31-s + 32-s + 0.962·33-s + 0.387·34-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 − 0.290·11-s − 0.288·12-s + 0.447·13-s + 0.895·14-s + 0.250·16-s + 0.0940·17-s + 0.235·18-s + 0.229·19-s − 0.731·21-s − 0.205·22-s + 0.200·23-s − 0.204·24-s + 0.316·26-s − 0.192·27-s + 0.633·28-s + 1.29·29-s + 0.601·31-s + 0.176·32-s + 0.167·33-s + 0.0665·34-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

\(L(1)\) \(\approx\) \(2.948551418\)
\(L(\frac12)\) \(\approx\) \(2.948551418\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.962T + 11T^{2} \)
13 \( 1 - 1.61T + 13T^{2} \)
17 \( 1 - 0.387T + 17T^{2} \)
23 \( 1 - 0.962T + 23T^{2} \)
29 \( 1 - 6.96T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 - 1.61T + 37T^{2} \)
41 \( 1 + 9.27T + 41T^{2} \)
43 \( 1 + 6.18T + 43T^{2} \)
47 \( 1 + 0.962T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 7.22T + 67T^{2} \)
71 \( 1 - 7.22T + 71T^{2} \)
73 \( 1 - 3.22T + 73T^{2} \)
79 \( 1 + 3.35T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 - 4.64T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−8.388894630390848685379926718386, −8.157547731331686439011018941587, −7.04385114077478658360195358489, −6.47082808670588982150472077549, −5.44789255013430167252404435544, −4.99160121048984436468009542127, −4.26807159652247261969913956316, −3.23510892995052702924737950843, −2.07330895612808480526761291781, −1.05493677737871007217103639281, 1.05493677737871007217103639281, 2.07330895612808480526761291781, 3.23510892995052702924737950843, 4.26807159652247261969913956316, 4.99160121048984436468009542127, 5.44789255013430167252404435544, 6.47082808670588982150472077549, 7.04385114077478658360195358489, 8.157547731331686439011018941587, 8.388894630390848685379926718386

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