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Properties

Label 2-2700-9.4-c1-0-15
Degree $2$
Conductor $2700$
Sign $-0.234 + 0.972i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
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  + (0.340 + 0.589i)7-s + (−0.840 − 1.45i)11-s + (2.57 − 4.45i)13-s − 1.31·17-s − 0.324·19-s + (−1.89 + 3.28i)23-s + (−4.32 − 7.48i)29-s + (2.07 − 3.58i)31-s − 1.35·37-s + (−3.57 + 6.19i)41-s + (−3.64 − 6.31i)43-s + (6.48 + 11.2i)47-s + (3.26 − 5.66i)49-s − 8.83·53-s + (4.40 − 7.63i)59-s + ⋯
L(s)  = 1  + (0.128 + 0.222i)7-s + (−0.253 − 0.438i)11-s + (0.713 − 1.23i)13-s − 0.320·17-s − 0.0744·19-s + (−0.395 + 0.684i)23-s + (−0.802 − 1.39i)29-s + (0.372 − 0.644i)31-s − 0.222·37-s + (−0.558 + 0.967i)41-s + (−0.555 − 0.962i)43-s + (0.945 + 1.63i)47-s + (0.466 − 0.808i)49-s − 1.21·53-s + (0.573 − 0.993i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.234 + 0.972i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.234 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.234373511\)
\(L(\frac12)\) \(\approx\) \(1.234373511\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.340 - 0.589i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.840 + 1.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.57 + 4.45i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 + 0.324T + 19T^{2} \)
23 \( 1 + (1.89 - 3.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.32 + 7.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.07 + 3.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.35T + 37T^{2} \)
41 \( 1 + (3.57 - 6.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.64 + 6.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.48 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.83T + 53T^{2} \)
59 \( 1 + (-4.40 + 7.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.98 + 8.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.08 - 3.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.891T + 71T^{2} \)
73 \( 1 - 7.82T + 73T^{2} \)
79 \( 1 + (4.82 + 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.42 + 4.20i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 + (-4.46 - 7.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.364068089054724299599877109925, −8.084256874623012125286406918486, −7.19387329956616588355522333925, −6.05045225412570013854515103720, −5.73820657875167720095842508258, −4.71234815778049209066645967258, −3.71521428400016916625700171820, −2.92239198765319028027625880196, −1.79663543066838015810877549354, −0.40163074616676264989797119049, 1.36587292864006128650683538780, 2.30255259313096482679343032264, 3.52999571760731213965198245145, 4.31229336303818546787397437118, 5.07793579336397721031920132231, 6.06663853269172108341987142238, 6.87389155640221616145672592181, 7.37208429196009917711949149475, 8.535297602527139546119536867704, 8.861803307951980336884145203416

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