Properties

Label 2-2550-5.4-c1-0-13
Degree $2$
Conductor $2550$
Sign $-0.447 - 0.894i$
Analytic cond. $20.3618$
Root an. cond. $4.51241$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s i·7-s i·8-s − 9-s − 3·11-s i·12-s − 4i·13-s + 14-s + 16-s i·17-s i·18-s + 5·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s − 0.904·11-s − 0.288i·12-s − 1.10i·13-s + 0.267·14-s + 0.250·16-s − 0.242i·17-s − 0.235i·18-s + 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 17\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(20.3618\)
Root analytic conductor: \(4.51241\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2550} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2550,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.377973371\)
\(L(\frac12)\) \(\approx\) \(1.377973371\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 \)
17 \( 1 + iT \)
good7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 7iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 15T + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−9.167407619902233046494222138902, −8.183956694854301605070786512292, −7.66821475429571238332984014363, −7.02104696732293284844404007740, −5.76003013289323366722328892513, −5.40710695070715015431231173032, −4.58256089390680482103131667670, −3.52363024777818826798405531636, −2.81238748262843188628141558196, −0.988243288867752142068473903686, 0.56614619185317674264079813213, 1.95033174118508858370398304990, 2.58568942522214367678035731820, 3.63809958627429807223981996121, 4.68388897049237560413442757728, 5.43143616327414868446114459656, 6.34975667134583752122398761345, 7.17862736801945245626673622924, 7.990158316841204488947389683822, 8.750665164041458623983831845462

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