Properties

Label 2-2850-1.1-c1-0-28
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 4.44·7-s − 8-s + 9-s + 3.44·11-s − 12-s + 2.44·13-s + 4.44·14-s + 16-s − 4.44·17-s − 18-s + 19-s + 4.44·21-s − 3.44·22-s + 23-s + 24-s − 2.44·26-s − 27-s − 4.44·28-s − 4.34·29-s − 3·31-s − 32-s − 3.44·33-s + 4.44·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.68·7-s − 0.353·8-s + 0.333·9-s + 1.04·11-s − 0.288·12-s + 0.679·13-s + 1.18·14-s + 0.250·16-s − 1.07·17-s − 0.235·18-s + 0.229·19-s + 0.970·21-s − 0.735·22-s + 0.208·23-s + 0.204·24-s − 0.480·26-s − 0.192·27-s − 0.840·28-s − 0.807·29-s − 0.538·31-s − 0.176·32-s − 0.600·33-s + 0.763·34-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: \(1\)
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 \( 1 + 4.44T + 7T^{2} \)
11 \( 1 - 3.44T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + 4.34T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + 0.898T + 41T^{2} \)
43 \( 1 - 2.44T + 43T^{2} \)
47 \( 1 - 7.79T + 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + 1.55T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 8.34T + 83T^{2} \)
89 \( 1 - 2.10T + 89T^{2} \)
97 \( 1 + 1.55T + 97T^{2} \)
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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

−8.751451936220509576523778186765, −7.43433276418775594442952202263, −6.88607281996833567728995304571, −6.18672088483711867071126449190, −5.76122180477598959985503379021, −4.28122472671208880741835202545, −3.58456769113566184992346954775, −2.52987735606926522428771591853, −1.19149362302376323593201201087, 0, 1.19149362302376323593201201087, 2.52987735606926522428771591853, 3.58456769113566184992346954775, 4.28122472671208880741835202545, 5.76122180477598959985503379021, 6.18672088483711867071126449190, 6.88607281996833567728995304571, 7.43433276418775594442952202263, 8.751451936220509576523778186765

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