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Properties

Label 2-2700-45.34-c1-0-7
Degree $2$
Conductor $2700$
Sign $0.558 - 0.829i$
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  + (3.55 + 2.05i)7-s + (−1.90 + 3.30i)11-s + (5.03 − 2.90i)13-s + 3.81i·17-s + 1.81·19-s + (−1.81 + 1.05i)23-s + (3.60 − 6.23i)29-s + (0.908 + 1.57i)31-s + 6.01i·37-s + (5.50 + 9.54i)41-s + (−5.03 − 2.90i)43-s + (−10.3 − 5.95i)47-s + (4.90 + 8.50i)49-s − 4.20i·53-s + (2.10 + 3.63i)59-s + ⋯
L(s)  = 1  + (1.34 + 0.774i)7-s + (−0.575 + 0.996i)11-s + (1.39 − 0.806i)13-s + 0.925i·17-s + 0.416·19-s + (−0.379 + 0.219i)23-s + (0.668 − 1.15i)29-s + (0.163 + 0.282i)31-s + 0.989i·37-s + (0.860 + 1.49i)41-s + (−0.768 − 0.443i)43-s + (−1.50 − 0.869i)47-s + (0.701 + 1.21i)49-s − 0.577i·53-s + (0.273 + 0.473i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ 0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.202070698\)
\(L(\frac12)\) \(\approx\) \(2.202070698\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.55 - 2.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.90 - 3.30i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.03 + 2.90i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.81iT - 17T^{2} \)
19 \( 1 - 1.81T + 19T^{2} \)
23 \( 1 + (1.81 - 1.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.60 + 6.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.908 - 1.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.01iT - 37T^{2} \)
41 \( 1 + (-5.50 - 9.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.03 + 2.90i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.3 + 5.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.20iT - 53T^{2} \)
59 \( 1 + (-2.10 - 3.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.50 + 2.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.21 - 1.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.01T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.37 + 1.94i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-10.5 - 6.10i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.634926767688510159681894450312, −8.208567333309837546377215665993, −7.77038284569704096063149452872, −6.56060868881354506553800755126, −5.80066106597264539181181886274, −5.07022608370953329227335325236, −4.35233308431947871252287992051, −3.24198776575385351692032866363, −2.13997050251469947477361517375, −1.30058401578037575612038471923, 0.804373857470996592668801108608, 1.75730095642661616619803048661, 3.06959249042814931751036180747, 3.98436563575764032833803631581, 4.78110144715440510519766482514, 5.55138953201827180958474142969, 6.44644810160186287769711597590, 7.31972421787039653276669533558, 8.013395962083558658415038839615, 8.616726682150812098462855207590

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