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Properties

Label 2-2550-5.4-c1-0-47
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s − 3i·7-s + i·8-s − 9-s − 5·11-s + i·12-s − 2i·13-s − 3·14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.13i·7-s + 0.353i·8-s − 0.333·9-s − 1.50·11-s + 0.288i·12-s − 0.554i·13-s − 0.801·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.229·19-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: \(1\)
Selberg data: \((2,\ 2550,\ (\ :1/2),\ 0.447 - 0.894i)\)

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 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 \)
17 \( 1 + iT \)
good7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 12iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−7.977376926333309118175734280291, −7.74454738505994152014273527758, −6.94274765063363805424984972340, −5.75379687540167972938753962772, −5.13243726669882680950299969767, −4.07853809484330157121327256064, −3.20650849535453936657773173674, −2.31404446576434806501420573190, −1.14469489677111027080542182623, 0, 2.15227897478228432824090523678, 2.98209019074261041681377611880, 4.21844450252741656683311538202, 4.92920505331257891477187201512, 5.71549906655029950639976440705, 6.20519285789021037069340056440, 7.34613799676728884530216814158, 8.062297722908699188937248350481, 8.753085849492145649410358642210

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