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Properties

Label 2-30e2-45.4-c1-0-2
Degree $2$
Conductor $900$
Sign $-0.232 - 0.972i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  − 1.73·3-s + (−0.866 + 0.5i)7-s + 2.99·9-s + (−1.5 − 2.59i)11-s + (−0.866 − 0.5i)13-s + 6i·17-s + 4·19-s + (1.49 − 0.866i)21-s + (−2.59 − 1.5i)23-s − 5.19·27-s + (1.5 + 2.59i)29-s + (−2.5 + 4.33i)31-s + (2.59 + 4.5i)33-s + 2i·37-s + (1.49 + 0.866i)39-s + ⋯
L(s)  = 1  − 1.00·3-s + (−0.327 + 0.188i)7-s + 0.999·9-s + (−0.452 − 0.783i)11-s + (−0.240 − 0.138i)13-s + 1.45i·17-s + 0.917·19-s + (0.327 − 0.188i)21-s + (−0.541 − 0.312i)23-s − 1.00·27-s + (0.278 + 0.482i)29-s + (−0.449 + 0.777i)31-s + (0.452 + 0.783i)33-s + 0.328i·37-s + (0.240 + 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.232 - 0.972i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.232 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391772 + 0.496380i\)
\(L(\frac12)\) \(\approx\) \(0.391772 + 0.496380i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.79 - 4.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.06 - 3.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.79 + 4.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-9.52 + 5.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.43443833144500461882960994279, −9.755238903304886813484416224161, −8.637311100814681188955652633364, −7.78031591159943851299602517631, −6.75191772929865634471874838465, −5.94694204763234731172117510110, −5.29855261790641708527844868228, −4.16987167615069017400765979649, −3.01706891644243722217438101505, −1.34188634750511963804654705086, 0.36360534282598208838367880784, 2.07002491690932786001713949038, 3.56140418230834929503433682170, 4.78633480532849259516871633903, 5.34033113431785909550557990415, 6.47948549080964466391781117707, 7.22050153103374247885806049551, 7.901326102532891935228629706379, 9.527238326989115526264805083961, 9.741010557528228785500709521499

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