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Properties

Label 2-2550-5.4-c1-0-24
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 − 3i·7-s + i·8-s − 9-s − 3·11-s − i·12-s + 4i·13-s − 3·14-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 − 1.13i·7-s + 0.353i·8-s − 0.333·9-s − 0.904·11-s − 0.288i·12-s + 1.10i·13-s − 0.801·14-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

\(L(1)\) \(\approx\) \(1.472512896\)
\(L(\frac12)\) \(\approx\) \(1.472512896\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + 3iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 11iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.063984090501083176599876361603, −7.994025721984093618081686782111, −7.41631173451711677372486432169, −6.46549599912781176754928565284, −5.29907367924317703658930503650, −4.69633561241729804880854703105, −3.84319716434560564264625713413, −3.14826157351925821323774196468, −1.97532343030252417689562816126, −0.65194847193031442096880134526, 0.910388054158702927031553283718, 2.49832566884053435184392029293, 3.08724985745249765195677137774, 4.55517624917281616818850059403, 5.47729904627234782254924807134, 5.81024039913179164477572307631, 6.72720772258649592744899043674, 7.62880002683200570807575205675, 8.172970612614319067779007494073, 8.711810076340587079370358538648

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