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Properties

Label 2-2700-9.4-c1-0-3
Degree $2$
Conductor $2700$
Sign $0.137 - 0.990i$
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  + (−2.49 − 4.32i)7-s + (1.99 + 3.46i)11-s + (0.771 − 1.33i)13-s − 6.99·17-s − 2.25·19-s + (−3.89 + 6.75i)23-s + (3.08 + 5.33i)29-s + (0.271 − 0.470i)31-s + 6.25·37-s + (0.0979 − 0.169i)41-s + (−0.0431 − 0.0747i)43-s + (1.91 + 3.31i)47-s + (−8.97 + 15.5i)49-s + 4.19·53-s + (3.51 − 6.08i)59-s + ⋯
L(s)  = 1  + (−0.944 − 1.63i)7-s + (0.602 + 1.04i)11-s + (0.213 − 0.370i)13-s − 1.69·17-s − 0.517·19-s + (−0.812 + 1.40i)23-s + (0.572 + 0.991i)29-s + (0.0487 − 0.0844i)31-s + 1.02·37-s + (0.0152 − 0.0264i)41-s + (−0.00658 − 0.0114i)43-s + (0.279 + 0.483i)47-s + (−1.28 + 2.22i)49-s + 0.575·53-s + (0.457 − 0.792i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.137 - 0.990i$
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: \((2,\ 2700,\ (\ :1/2),\ 0.137 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8562637766\)
\(L(\frac12)\) \(\approx\) \(0.8562637766\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 + 4.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.99 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.771 + 1.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.99T + 17T^{2} \)
19 \( 1 + 2.25T + 19T^{2} \)
23 \( 1 + (3.89 - 6.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.08 - 5.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.271 + 0.470i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.25T + 37T^{2} \)
41 \( 1 + (-0.0979 + 0.169i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0431 + 0.0747i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.91 - 3.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.19T + 53T^{2} \)
59 \( 1 + (-3.51 + 6.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.45 - 2.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.48 + 7.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 - 2.28T + 73T^{2} \)
79 \( 1 + (-6.32 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.98 - 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (-4.66 - 8.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.279810001149367271879783153564, −8.133603286645851923943147037144, −7.35750808466117897849857814503, −6.73132414817612216394932104200, −6.26516004675440416231830378686, −4.92428999587726058145466731026, −4.09232291787388657322323951946, −3.64261934060268922481905227070, −2.32446836611097229554104711968, −1.09331883629882209740052205234, 0.30238215803823744485330269869, 2.16840228093825969995445582415, 2.71102493796635944869142657226, 3.86178926020645931022489251250, 4.66868499542958691153667999092, 6.00559887002682462076214887480, 6.14341207468134505582758214309, 6.84705519586216884443890629404, 8.289052891619699559046944690722, 8.725017949912561781566753858535

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