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Properties

Label 2-2550-5.4-c1-0-4
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 + 4i·7-s + i·8-s − 9-s + 2·11-s + i·12-s − 2i·13-s + 4·14-s + 16-s − i·17-s + i·18-s − 8·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.51i·7-s + 0.353i·8-s − 0.333·9-s + 0.603·11-s + 0.288i·12-s − 0.554i·13-s + 1.06·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 1.83·19-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

\(L(1)\) \(\approx\) \(0.8073285463\)
\(L(\frac12)\) \(\approx\) \(0.8073285463\)
\(L(\frac{3}{2})\) 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 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 13iT - 53T^{2} \)
59 \( 1 + 15T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.914147617683957278269128815251, −8.513266821007712667320951346169, −7.72210094517840288875288565481, −6.47707174733260893400457293623, −6.04916120122282390631626381735, −5.08735119759958399972964687477, −4.20433673111469906716071038403, −2.95063153621646548487442655277, −2.38480716153883640442766421381, −1.34996548618509835653470766168, 0.27030444657839510129600609241, 1.78058518827819546049342350544, 3.42797119317678671543979844038, 4.20069463002536196815285492343, 4.55627179694115357332180529950, 5.74148591225785876529250480653, 6.65404525500654111465341399049, 7.02716953861322278269096910719, 8.039339131730361769055049848285, 8.699014297837687460777474835280

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