Properties

Label 2-2550-1.1-c1-0-19
Degree $2$
Conductor $2550$
Sign $1$
Analytic cond. $20.3618$
Root an. cond. $4.51241$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 4·7-s − 8-s + 9-s − 2·11-s + 12-s + 6·13-s − 4·14-s + 16-s + 17-s − 18-s + 4·19-s + 4·21-s + 2·22-s − 5·23-s − 24-s − 6·26-s + 27-s + 4·28-s + 10·31-s − 32-s − 2·33-s − 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.426·22-s − 1.04·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s + 0.755·28-s + 1.79·31-s − 0.176·32-s − 0.348·33-s − 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(20.3618\)
Root analytic conductor: \(4.51241\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.147351202\)
\(L(\frac12)\) \(\approx\) \(2.147351202\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.605637047293088628845868985582, −8.188287204909209328453849372785, −7.80740081157824912500538828494, −6.79463681343707149067236637719, −5.84084108193046638467245630585, −4.99585376514666012367397104697, −3.99225334470857281631445650152, −3.01308414951224896614167330334, −1.88247361842834831005657001491, −1.11251462700235043615266083539, 1.11251462700235043615266083539, 1.88247361842834831005657001491, 3.01308414951224896614167330334, 3.99225334470857281631445650152, 4.99585376514666012367397104697, 5.84084108193046638467245630585, 6.79463681343707149067236637719, 7.80740081157824912500538828494, 8.188287204909209328453849372785, 8.605637047293088628845868985582

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