Properties

Label 2-2550-1.1-c1-0-44
Degree $2$
Conductor $2550$
Sign $-1$
Analytic cond. $20.3618$
Root an. cond. $4.51241$
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 − 2.44·7-s + 8-s + 9-s + 4.89·11-s − 12-s − 6·13-s − 2.44·14-s + 16-s − 17-s + 18-s − 2.89·19-s + 2.44·21-s + 4.89·22-s − 1.55·23-s − 24-s − 6·26-s − 27-s − 2.44·28-s + 5.34·29-s − 1.55·31-s + 32-s − 4.89·33-s − 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.925·7-s + 0.353·8-s + 0.333·9-s + 1.47·11-s − 0.288·12-s − 1.66·13-s − 0.654·14-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.665·19-s + 0.534·21-s + 1.04·22-s − 0.323·23-s − 0.204·24-s − 1.17·26-s − 0.192·27-s − 0.462·28-s + 0.993·29-s − 0.278·31-s + 0.176·32-s − 0.852·33-s − 0.171·34-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 + T \)
good7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
23 \( 1 + 1.55T + 23T^{2} \)
29 \( 1 - 5.34T + 29T^{2} \)
31 \( 1 + 1.55T + 31T^{2} \)
37 \( 1 + 4.44T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 6.89T + 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 + 1.34T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 + 5.10T + 73T^{2} \)
79 \( 1 + 6.44T + 79T^{2} \)
83 \( 1 - 6.89T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.580146443321086626411745454568, −7.37949091513163869652225485929, −6.69184351502152784990078135798, −6.36345867043676123948464107705, −5.32107224286238112333698688631, −4.55708370717111953737147648800, −3.79062353951754526879715451853, −2.81354291612170889049388687258, −1.66325318979744769235207780364, 0, 1.66325318979744769235207780364, 2.81354291612170889049388687258, 3.79062353951754526879715451853, 4.55708370717111953737147648800, 5.32107224286238112333698688631, 6.36345867043676123948464107705, 6.69184351502152784990078135798, 7.37949091513163869652225485929, 8.580146443321086626411745454568

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