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Properties

Label 2-2700-9.2-c2-0-14
Degree $2$
Conductor $2700$
Sign $-0.118 - 0.993i$
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  + (−4.05 + 7.02i)7-s + (17.6 + 10.1i)11-s + (3.05 + 5.29i)13-s + 17.9i·17-s + 9.11·19-s + (29.0 − 16.7i)23-s + (−14.4 − 8.31i)29-s + (11.1 + 19.3i)31-s + 50.4·37-s + (−29.9 + 17.3i)41-s + (11.5 − 19.9i)43-s + (−33.1 − 19.1i)47-s + (−8.44 − 14.6i)49-s + 19.0i·53-s + (2.96 − 1.71i)59-s + ⋯
L(s)  = 1  + (−0.579 + 1.00i)7-s + (1.60 + 0.924i)11-s + (0.235 + 0.407i)13-s + 1.05i·17-s + 0.479·19-s + (1.26 − 0.729i)23-s + (−0.496 − 0.286i)29-s + (0.360 + 0.624i)31-s + 1.36·37-s + (−0.730 + 0.421i)41-s + (0.267 − 0.463i)43-s + (−0.705 − 0.407i)47-s + (−0.172 − 0.298i)49-s + 0.358i·53-s + (0.0502 − 0.0290i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.118 - 0.993i$
Analytic conductor: \(73.5696\)
Root analytic conductor: \(8.57727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1),\ -0.118 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.179757696\)
\(L(\frac12)\) \(\approx\) \(2.179757696\)
\(L(2)\) 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 \( 1 + (4.05 - 7.02i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-17.6 - 10.1i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.05 - 5.29i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 17.9iT - 289T^{2} \)
19 \( 1 - 9.11T + 361T^{2} \)
23 \( 1 + (-29.0 + 16.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (14.4 + 8.31i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 50.4T + 1.36e3T^{2} \)
41 \( 1 + (29.9 - 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.5 + 19.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (33.1 + 19.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 19.0iT - 2.80e3T^{2} \)
59 \( 1 + (-2.96 + 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 + 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.14 + 5.45i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 35.9iT - 5.04e3T^{2} \)
73 \( 1 + 47.3T + 5.32e3T^{2} \)
79 \( 1 + (-42.2 + 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (33.1 + 19.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (-40.3 + 69.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.989615130046499653172944828342, −8.294593635350579952991864800544, −7.16142304108677323918671078243, −6.51817278293167616861627513243, −5.99174256684897067021011581190, −4.91054884462892774810071553271, −4.09241743532611627614050114811, −3.21700364719051505513062360834, −2.14222242030574067053125145265, −1.18715584555529499073155070522, 0.59992131537429801932572073052, 1.27326835958508384156676099268, 2.95993246768974988663692348397, 3.55765610120644626613567086932, 4.35785351971496895510349870631, 5.41812332879837926262158869420, 6.26664580823447574189933999111, 6.95469517580251247690486703557, 7.54228365129640928319297341428, 8.536722125650262063896636906573

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