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Properties

Label 2-2700-45.29-c2-0-31
Degree $2$
Conductor $2700$
Sign $-0.00372 + 0.999i$
Analytic cond. $73.5696$
Root an. cond. $8.57727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.89 + 4.55i)7-s + (0.383 + 0.221i)11-s + (−9.62 + 5.55i)13-s − 8.01·17-s + 8.11·19-s + (−11.8 − 20.4i)23-s + (−45.9 − 26.5i)29-s + (−14.6 − 25.4i)31-s − 18.4i·37-s + (38.9 − 22.4i)41-s + (−19.9 − 11.5i)43-s + (4.22 − 7.32i)47-s + (17.0 + 29.5i)49-s − 60.5·53-s + (65.9 − 38.0i)59-s + ⋯
L(s)  = 1  + (1.12 + 0.651i)7-s + (0.0348 + 0.0201i)11-s + (−0.740 + 0.427i)13-s − 0.471·17-s + 0.427·19-s + (−0.513 − 0.888i)23-s + (−1.58 − 0.913i)29-s + (−0.473 − 0.819i)31-s − 0.499i·37-s + (0.950 − 0.548i)41-s + (−0.463 − 0.267i)43-s + (0.0899 − 0.155i)47-s + (0.348 + 0.602i)49-s − 1.14·53-s + (1.11 − 0.645i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00372 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00372 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.00372 + 0.999i$
Analytic conductor: \(73.5696\)
Root analytic conductor: \(8.57727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (2249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1),\ -0.00372 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.369925540\)
\(L(\frac12)\) \(\approx\) \(1.369925540\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-7.89 - 4.55i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.383 - 0.221i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.62 - 5.55i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 8.01T + 289T^{2} \)
19 \( 1 - 8.11T + 361T^{2} \)
23 \( 1 + (11.8 + 20.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (45.9 + 26.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.6 + 25.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 18.4iT - 1.36e3T^{2} \)
41 \( 1 + (-38.9 + 22.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.9 + 11.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-4.22 + 7.32i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 60.5T + 2.80e3T^{2} \)
59 \( 1 + (-65.9 + 38.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.67 - 4.63i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-95.0 + 54.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 16.0iT - 5.04e3T^{2} \)
73 \( 1 + 4.35iT - 5.32e3T^{2} \)
79 \( 1 + (-0.792 + 1.37i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (4.22 - 7.32i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 64.1iT - 7.92e3T^{2} \)
97 \( 1 + (-99.7 - 57.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−8.404194099774197994678935706326, −7.78390145507489982725167735016, −7.07258742804866585776771976270, −6.05176204062072677799797937439, −5.33849774485079058028754540386, −4.57965624873638936304070978176, −3.78108719468679946462175590011, −2.34534139538323604810331791533, −1.91729106091421642986434671035, −0.31830834830109825940385016031, 1.14022861469020228179360083138, 2.02280971895607046122147940781, 3.24422349821869015758487660968, 4.14304758596364379345397111720, 5.01268297135663945437333995442, 5.54872448762959939034292965246, 6.72427197582153266345368388176, 7.51391706868310235791713729917, 7.88794307660903938693860044499, 8.829064655972105700318218850793

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