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Properties

Label 2-30e2-5.3-c2-0-3
Degree $2$
Conductor $900$
Sign $0.437 - 0.899i$
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  + (3.67 − 3.67i)7-s − 6·11-s + (6.12 + 6.12i)13-s + (−22.0 + 22.0i)17-s + 25i·19-s + (7.34 + 7.34i)23-s − 42i·29-s + 49·31-s + (4.89 − 4.89i)37-s + 60·41-s + (−1.22 − 1.22i)43-s + (−51.4 + 51.4i)47-s + 22i·49-s + (14.6 + 14.6i)53-s + 78i·59-s + ⋯
L(s)  = 1  + (0.524 − 0.524i)7-s − 0.545·11-s + (0.471 + 0.471i)13-s + (−1.29 + 1.29i)17-s + 1.31i·19-s + (0.319 + 0.319i)23-s − 1.44i·29-s + 1.58·31-s + (0.132 − 0.132i)37-s + 1.46·41-s + (−0.0284 − 0.0284i)43-s + (−1.09 + 1.09i)47-s + 0.448i·49-s + (0.277 + 0.277i)53-s + 1.32i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.437 - 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.638816893\)
\(L(\frac12)\) \(\approx\) \(1.638816893\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-3.67 + 3.67i)T - 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (-6.12 - 6.12i)T + 169iT^{2} \)
17 \( 1 + (22.0 - 22.0i)T - 289iT^{2} \)
19 \( 1 - 25iT - 361T^{2} \)
23 \( 1 + (-7.34 - 7.34i)T + 529iT^{2} \)
29 \( 1 + 42iT - 841T^{2} \)
31 \( 1 - 49T + 961T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 1.36e3iT^{2} \)
41 \( 1 - 60T + 1.68e3T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + 1.84e3iT^{2} \)
47 \( 1 + (51.4 - 51.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (-14.6 - 14.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 78iT - 3.48e3T^{2} \)
61 \( 1 + 13T + 3.72e3T^{2} \)
67 \( 1 + (52.6 - 52.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 60T + 5.04e3T^{2} \)
73 \( 1 + (-63.6 - 63.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 106iT - 6.24e3T^{2} \)
83 \( 1 + (-80.8 - 80.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 60iT - 7.92e3T^{2} \)
97 \( 1 + (121. - 121. i)T - 9.40e3iT^{2} \)
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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

−10.19756134627435079034860463130, −9.214887415928290388343446730142, −8.189103346515548943543778654422, −7.77584717476103582678046483592, −6.50163370192797331852889148937, −5.87307579789237154076979584999, −4.51168124067335082404485464141, −3.95598329170529753294051297315, −2.44769502010000732833870298016, −1.25246216476932539739697648173, 0.57667412700053492618667834071, 2.24737243699608820486550056952, 3.11194699763948654194938815227, 4.71345348897573792484976655758, 5.11386515279209478711894010017, 6.40514148579235707583684229085, 7.15421053719414401496838464025, 8.237801470485185908074691259924, 8.852497858189045752543764093984, 9.658389161006736432612782977227

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