Properties

Label 2-2550-1.1-c1-0-33
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s − 2·13-s + 16-s − 17-s − 18-s − 4·19-s − 4·22-s − 4·23-s + 24-s + 2·26-s − 27-s + 2·29-s − 4·31-s − 32-s − 4·33-s + 34-s + 36-s + 6·37-s + 4·38-s + 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + 0.171·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-s + 0.320·39-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.508076357926522304587707936931, −7.85391852680015313417178593889, −6.81293701478766628611331020079, −6.47749789975439920149959740273, −5.55607683757389846715119248044, −4.51444126045046681585118943777, −3.71442293834107121729389823605, −2.36951231460413935519804036920, −1.37551986425423722963512172703, 0, 1.37551986425423722963512172703, 2.36951231460413935519804036920, 3.71442293834107121729389823605, 4.51444126045046681585118943777, 5.55607683757389846715119248044, 6.47749789975439920149959740273, 6.81293701478766628611331020079, 7.85391852680015313417178593889, 8.508076357926522304587707936931

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