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Properties

Label 2-30e2-4.3-c2-0-28
Degree $2$
Conductor $900$
Sign $-0.5 - 0.866i$
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 + 1.73i)2-s + (−1.99 − 3.46i)4-s + 10.3i·7-s + 7.99·8-s − 10.3i·11-s + 18·13-s + (−18 − 10.3i)14-s + (−8 + 13.8i)16-s − 10·17-s + 13.8i·19-s + (18 + 10.3i)22-s − 6.92i·23-s + (−18 + 31.1i)26-s + (36 − 20.7i)28-s + 36·29-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.48i·7-s + 0.999·8-s − 0.944i·11-s + 1.38·13-s + (−1.28 − 0.742i)14-s + (−0.5 + 0.866i)16-s − 0.588·17-s + 0.729i·19-s + (0.818 + 0.472i)22-s − 0.301i·23-s + (−0.692 + 1.19i)26-s + (1.28 − 0.742i)28-s + 1.24·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.5 - 0.866i$
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: \((2,\ 900,\ (\ :1),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.306898345\)
\(L(\frac12)\) \(\approx\) \(1.306898345\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 10.3iT - 49T^{2} \)
11 \( 1 + 10.3iT - 121T^{2} \)
13 \( 1 - 18T + 169T^{2} \)
17 \( 1 + 10T + 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 + 6.92iT - 529T^{2} \)
29 \( 1 - 36T + 841T^{2} \)
31 \( 1 + 6.92iT - 961T^{2} \)
37 \( 1 - 54T + 1.36e3T^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 - 20.7iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 26T + 2.80e3T^{2} \)
59 \( 1 - 31.1iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 - 41.5iT - 4.48e3T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 - 36T + 5.32e3T^{2} \)
79 \( 1 - 90.0iT - 6.24e3T^{2} \)
83 \( 1 - 90.0iT - 6.88e3T^{2} \)
89 \( 1 - 18T + 7.92e3T^{2} \)
97 \( 1 + 72T + 9.40e3T^{2} \)
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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

−9.981751637693959073978559135186, −8.990590261134404974295501635210, −8.551912277390231074184078548440, −7.954042425382920447742641694216, −6.48519449670707884957183400033, −6.06359477420509383610556088561, −5.30212225250317417093816116493, −4.05260414601340605292945754260, −2.61787344751079124540848566845, −1.13885131245011505877905370491, 0.61535999255230883171405874797, 1.69064483572484304016050497037, 3.11534711763170037412413876757, 4.13001374729030913117272241390, 4.71905351340388600562903938607, 6.45394860019986659299713521235, 7.25043517686465679580226277914, 8.042357073702417448344540208087, 8.953721199796506089605513759842, 9.773719138048482954677431317244

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