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Properties

Label 2-30e2-4.3-c2-0-42
Degree $2$
Conductor $900$
Sign $0.798 - 0.602i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.89 − 0.635i)2-s + (3.19 − 2.40i)4-s + 10.1i·7-s + (4.52 − 6.59i)8-s + 10.6i·11-s + 11.7·13-s + (6.41 + 19.1i)14-s + (4.38 − 15.3i)16-s − 16.6·17-s + 0.464i·19-s + (6.77 + 20.2i)22-s + 42.1i·23-s + (22.3 − 7.47i)26-s + (24.3 + 32.2i)28-s + 19.5·29-s + ⋯
L(s)  = 1  + (0.948 − 0.317i)2-s + (0.798 − 0.602i)4-s + 1.44i·7-s + (0.565 − 0.824i)8-s + 0.968i·11-s + 0.905·13-s + (0.458 + 1.36i)14-s + (0.274 − 0.961i)16-s − 0.978·17-s + 0.0244i·19-s + (0.307 + 0.918i)22-s + 1.83i·23-s + (0.858 − 0.287i)26-s + (0.869 + 1.15i)28-s + 0.674·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.798 - 0.602i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.798 - 0.602i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.440890038\)
\(L(\frac12)\) \(\approx\) \(3.440890038\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.89 + 0.635i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 10.1iT - 49T^{2} \)
11 \( 1 - 10.6iT - 121T^{2} \)
13 \( 1 - 11.7T + 169T^{2} \)
17 \( 1 + 16.6T + 289T^{2} \)
19 \( 1 - 0.464iT - 361T^{2} \)
23 \( 1 - 42.1iT - 529T^{2} \)
29 \( 1 - 19.5T + 841T^{2} \)
31 \( 1 - 9.17iT - 961T^{2} \)
37 \( 1 - 23.5T + 1.36e3T^{2} \)
41 \( 1 + 44.0T + 1.68e3T^{2} \)
43 \( 1 - 58.3iT - 1.84e3T^{2} \)
47 \( 1 + 41.6iT - 2.20e3T^{2} \)
53 \( 1 - 80.2T + 2.80e3T^{2} \)
59 \( 1 - 63.9iT - 3.48e3T^{2} \)
61 \( 1 - 80.3T + 3.72e3T^{2} \)
67 \( 1 + 22.5iT - 4.48e3T^{2} \)
71 \( 1 + 61.4iT - 5.04e3T^{2} \)
73 \( 1 - 137.T + 5.32e3T^{2} \)
79 \( 1 + 138. iT - 6.24e3T^{2} \)
83 \( 1 + 86.2iT - 6.88e3T^{2} \)
89 \( 1 + 127.T + 7.92e3T^{2} \)
97 \( 1 - 15T + 9.40e3T^{2} \)
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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

−10.05314588224669071272431124209, −9.288346490101804904699425497165, −8.391542463445111599198569306463, −7.19809400646666822137311784948, −6.31651360160713232090048889523, −5.53945093646820085265127594414, −4.73123492045236014265440420611, −3.62309089879754819386187331242, −2.52052271892666413380279435549, −1.60523138485116039136992026409, 0.829931552898477843579040396499, 2.51546020332425827022788185318, 3.76456651514881417511218247611, 4.26932164777529739960315018490, 5.40888713416559579269613629720, 6.57508188484062204658994110337, 6.84175418188419595079063692262, 8.149237708776018757703809722306, 8.594251996233717638794345962813, 10.13474466432236914222791225197

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