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Properties

Label 2-30e2-25.21-c1-0-1
Degree $2$
Conductor $900$
Sign $-0.985 - 0.166i$
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  + (−2.04 + 0.909i)5-s + 0.747·7-s + (−0.0646 + 0.198i)11-s + (−0.773 − 2.38i)13-s + (−5.51 + 4.00i)17-s + (−1.00 + 0.731i)19-s + (−1.00 + 3.09i)23-s + (3.34 − 3.71i)25-s + (−4.19 − 3.04i)29-s + (−3.02 + 2.19i)31-s + (−1.52 + 0.679i)35-s + (0.607 + 1.86i)37-s + (−0.993 − 3.05i)41-s − 12.7·43-s + (−5.24 − 3.81i)47-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)5-s + 0.282·7-s + (−0.0194 + 0.0599i)11-s + (−0.214 − 0.660i)13-s + (−1.33 + 0.972i)17-s + (−0.231 + 0.167i)19-s + (−0.209 + 0.644i)23-s + (0.669 − 0.743i)25-s + (−0.778 − 0.565i)29-s + (−0.543 + 0.394i)31-s + (−0.258 + 0.114i)35-s + (0.0998 + 0.307i)37-s + (−0.155 − 0.477i)41-s − 1.93·43-s + (−0.764 − 0.555i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.985 - 0.166i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.985 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0196935 + 0.234523i\)
\(L(\frac12)\) \(\approx\) \(0.0196935 + 0.234523i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 - 0.909i)T \)
good7 \( 1 - 0.747T + 7T^{2} \)
11 \( 1 + (0.0646 - 0.198i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.773 + 2.38i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.51 - 4.00i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.00 - 0.731i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.00 - 3.09i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.19 + 3.04i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.02 - 2.19i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.607 - 1.86i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.993 + 3.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 + (5.24 + 3.81i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.35 - 2.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.61 + 11.1i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.85 - 11.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.35 - 1.71i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-5.29 - 3.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.778 - 2.39i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.28 - 6.02i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.59 - 3.33i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.284 + 0.876i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-12.5 - 9.13i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.64251933437197901917481405555, −9.767991627813993153338014393679, −8.585656801275537480300831195890, −8.064434198977441282785623288063, −7.15456659103349991659026771880, −6.33511511893711623327877106589, −5.17229331285673122501733282949, −4.15320490367894111535401487088, −3.30208626163912185911476316605, −1.92349917869334029384431190505, 0.10644550047817944334038898810, 1.92159222330698007856009420601, 3.32342202250942430923879533545, 4.47948736470954417482083258674, 4.97637796098290115089565161816, 6.42922210304949620435592018875, 7.19184497287617707978090650541, 8.063105259376520058180356706459, 8.882009258752647734685124165722, 9.500659663658012691412874044793

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