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Properties

Label 2-2700-9.7-c1-0-9
Degree $2$
Conductor $2700$
Sign $0.978 - 0.204i$
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  + (0.0432 − 0.0748i)7-s + (−0.456 + 0.791i)11-s + (1.31 + 2.27i)13-s + 2.08·17-s + 4.93·19-s + (−4.23 − 7.34i)23-s + (1.19 − 2.07i)29-s + (−1.81 − 3.13i)31-s + 5.85·37-s + (3.32 + 5.75i)41-s + (−4.12 + 7.14i)43-s + (−1.34 + 2.32i)47-s + (3.49 + 6.05i)49-s − 5.73·53-s + (6.16 + 10.6i)59-s + ⋯
L(s)  = 1  + (0.0163 − 0.0283i)7-s + (−0.137 + 0.238i)11-s + (0.364 + 0.630i)13-s + 0.506·17-s + 1.13·19-s + (−0.883 − 1.53i)23-s + (0.222 − 0.385i)29-s + (−0.325 − 0.563i)31-s + 0.962·37-s + (0.519 + 0.899i)41-s + (−0.629 + 1.08i)43-s + (−0.196 + 0.339i)47-s + (0.499 + 0.865i)49-s − 0.787·53-s + (0.803 + 1.39i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.978 - 0.204i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ 0.978 - 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.857975501\)
\(L(\frac12)\) \(\approx\) \(1.857975501\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0432 + 0.0748i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.456 - 0.791i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.31 - 2.27i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.08T + 17T^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 + (4.23 + 7.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.19 + 2.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.81 + 3.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.85T + 37T^{2} \)
41 \( 1 + (-3.32 - 5.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.12 - 7.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.34 - 2.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.73T + 53T^{2} \)
59 \( 1 + (-6.16 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.16 + 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.08 - 5.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 5.31T + 73T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.03 + 5.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.13T + 89T^{2} \)
97 \( 1 + (-5.55 + 9.61i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.878849539264768158947932634430, −7.998309989203068561250294950500, −7.50531662753454874612711597588, −6.43259177686133762664173683844, −5.94950261978184051265381794460, −4.83012428482304110634589430130, −4.19234799577826629138751811464, −3.13816911993635850489937468544, −2.17089872541339936092646595611, −0.918839113001161058882514654039, 0.826891421060996421556660291726, 2.03560395862346963406110539352, 3.30584020167770635244076078587, 3.77116670235263463363994536967, 5.22897504230761147504360784754, 5.49802638683358082943003420878, 6.52135757400341952068227447228, 7.44397775077159487645697492100, 7.976044831574804985528033270298, 8.778245495012415817539381016511

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