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Properties

Label 2-30e2-25.4-c1-0-0
Degree $2$
Conductor $900$
Sign $-0.997 - 0.0763i$
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  + (−1.99 + 1.00i)5-s + 3.80i·7-s + (−0.0589 + 0.181i)11-s + (1.59 − 0.518i)13-s + (−2.70 − 3.72i)17-s + (−2.13 + 1.55i)19-s + (−6.04 − 1.96i)23-s + (2.99 − 4.00i)25-s + (−2.03 − 1.48i)29-s + (−3.03 + 2.20i)31-s + (−3.81 − 7.61i)35-s + (−11.2 + 3.66i)37-s + (2.22 + 6.83i)41-s − 9.22i·43-s + (−2.67 + 3.67i)47-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s + 1.44i·7-s + (−0.0177 + 0.0546i)11-s + (0.442 − 0.143i)13-s + (−0.656 − 0.903i)17-s + (−0.490 + 0.356i)19-s + (−1.26 − 0.409i)23-s + (0.598 − 0.800i)25-s + (−0.378 − 0.275i)29-s + (−0.544 + 0.395i)31-s + (−0.645 − 1.28i)35-s + (−1.85 + 0.602i)37-s + (0.346 + 1.06i)41-s − 1.40i·43-s + (−0.389 + 0.536i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.997 - 0.0763i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.997 - 0.0763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0163580 + 0.427808i\)
\(L(\frac12)\) \(\approx\) \(0.0163580 + 0.427808i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.99 - 1.00i)T \)
good7 \( 1 - 3.80iT - 7T^{2} \)
11 \( 1 + (0.0589 - 0.181i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.59 + 0.518i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.70 + 3.72i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.13 - 1.55i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (6.04 + 1.96i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.03 + 1.48i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.03 - 2.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (11.2 - 3.66i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.22 - 6.83i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.22iT - 43T^{2} \)
47 \( 1 + (2.67 - 3.67i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.54 - 7.62i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.20 - 6.79i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.94 + 9.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.55 - 4.89i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-10.7 - 7.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.95 + 1.61i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.51 + 1.82i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.74 - 3.78i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.30 + 13.2i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.93 - 5.41i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.64126718467086195988685924881, −9.615911293955678494966894128650, −8.661012359120750376164461890185, −8.196416695822384346713428041215, −7.09272978625994046367012058466, −6.24657366193463015418671303798, −5.31322485618641940136909602059, −4.21417135659211153042820072703, −3.12767886742380912201477208986, −2.10477001618968295738541157720, 0.19714745501967071925020296465, 1.72567170694285694104532264552, 3.72991901185139290060280042626, 3.98195584953158730932383280131, 5.12345729791357241659447236569, 6.42203269438556598702590076665, 7.22410182231061472844454668660, 8.015370116051220984988039670096, 8.704446501124078847796278146957, 9.742603459720887940349535271400

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