Properties

Label 2-30e2-5.2-c2-0-7
Degree $2$
Conductor $900$
Sign $0.793 - 0.608i$
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  + (4.89 + 4.89i)7-s + 15·11-s + (2.44 − 2.44i)13-s + (−3.67 − 3.67i)17-s + 17i·19-s + (−7.34 + 7.34i)23-s − 42i·29-s − 14·31-s + (29.3 + 29.3i)37-s + 39·41-s + (−24.4 + 24.4i)43-s + (51.4 + 51.4i)47-s − 1.00i·49-s + (−66.1 + 66.1i)53-s + 6i·59-s + ⋯
L(s)  = 1  + (0.699 + 0.699i)7-s + 1.36·11-s + (0.188 − 0.188i)13-s + (−0.216 − 0.216i)17-s + 0.894i·19-s + (−0.319 + 0.319i)23-s − 1.44i·29-s − 0.451·31-s + (0.794 + 0.794i)37-s + 0.951·41-s + (−0.569 + 0.569i)43-s + (1.09 + 1.09i)47-s − 0.0204i·49-s + (−1.24 + 1.24i)53-s + 0.101i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.793 - 0.608i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.793 - 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.204530817\)
\(L(\frac12)\) \(\approx\) \(2.204530817\)
\(L(2)\) 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 \( 1 + (-4.89 - 4.89i)T + 49iT^{2} \)
11 \( 1 - 15T + 121T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 169iT^{2} \)
17 \( 1 + (3.67 + 3.67i)T + 289iT^{2} \)
19 \( 1 - 17iT - 361T^{2} \)
23 \( 1 + (7.34 - 7.34i)T - 529iT^{2} \)
29 \( 1 + 42iT - 841T^{2} \)
31 \( 1 + 14T + 961T^{2} \)
37 \( 1 + (-29.3 - 29.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 39T + 1.68e3T^{2} \)
43 \( 1 + (24.4 - 24.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-51.4 - 51.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (66.1 - 66.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 6iT - 3.48e3T^{2} \)
61 \( 1 - 92T + 3.72e3T^{2} \)
67 \( 1 + (-35.5 - 35.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 102T + 5.04e3T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 5.32e3iT^{2} \)
79 \( 1 + 104iT - 6.24e3T^{2} \)
83 \( 1 + (-47.7 + 47.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 87iT - 7.92e3T^{2} \)
97 \( 1 + (93.0 + 93.0i)T + 9.40e3iT^{2} \)
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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

−9.813146977053839563942957970044, −9.215905094242959891176417630522, −8.301649415075825223954750326008, −7.62389671704156941582194167691, −6.37808573395805099684098923236, −5.78772362522137120389039790273, −4.60509981864677356366174900992, −3.73453364050487988786542271299, −2.35068377413279364373578174542, −1.20571028495290870933754448896, 0.863874214701753861927612612030, 2.03393110072668794765628187739, 3.62508188381631494354822269326, 4.34848410290523180983354929119, 5.36723735178860636615476174926, 6.60134702868559508699228131762, 7.12518961296624531123614457166, 8.218693997594543488716422246089, 8.987518952070984831642539660483, 9.736556748817604125340039394739

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