Properties

Label 2-30e2-9.2-c2-0-11
Degree $2$
Conductor $900$
Sign $0.219 - 0.975i$
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  + (2.91 − 0.708i)3-s + (−4.96 + 8.59i)7-s + (7.99 − 4.12i)9-s + (3.59 + 2.07i)11-s + (−0.613 − 1.06i)13-s + 21.0i·17-s − 9.84·19-s + (−8.38 + 28.5i)21-s + (2.32 − 1.34i)23-s + (20.3 − 17.6i)27-s + (−0.321 − 0.185i)29-s + (22.1 + 38.4i)31-s + (11.9 + 3.50i)33-s − 33.0·37-s + (−2.53 − 2.66i)39-s + ⋯
L(s)  = 1  + (0.971 − 0.236i)3-s + (−0.708 + 1.22i)7-s + (0.888 − 0.458i)9-s + (0.326 + 0.188i)11-s + (−0.0471 − 0.0817i)13-s + 1.23i·17-s − 0.517·19-s + (−0.399 + 1.36i)21-s + (0.100 − 0.0582i)23-s + (0.755 − 0.655i)27-s + (−0.0110 − 0.00640i)29-s + (0.715 + 1.23i)31-s + (0.361 + 0.106i)33-s − 0.893·37-s + (−0.0651 − 0.0682i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.219 - 0.975i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.219 - 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.193931952\)
\(L(\frac12)\) \(\approx\) \(2.193931952\)
\(L(2)\) 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 \( 1 + (-2.91 + 0.708i)T \)
5 \( 1 \)
good7 \( 1 + (4.96 - 8.59i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.59 - 2.07i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.613 + 1.06i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 21.0iT - 289T^{2} \)
19 \( 1 + 9.84T + 361T^{2} \)
23 \( 1 + (-2.32 + 1.34i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (0.321 + 0.185i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-22.1 - 38.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 33.0T + 1.36e3T^{2} \)
41 \( 1 + (34.2 - 19.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.5 - 25.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-68.6 - 39.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 92.8iT - 2.80e3T^{2} \)
59 \( 1 + (53.3 - 30.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12.6 + 21.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.9 - 31.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 16.7iT - 5.04e3T^{2} \)
73 \( 1 - 62.4T + 5.32e3T^{2} \)
79 \( 1 + (-71.9 + 124. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (14.2 + 8.20i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 155. iT - 7.92e3T^{2} \)
97 \( 1 + (52.3 - 90.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.948758205942813086803017542049, −8.976139428235741157908574128668, −8.677767486354693376630458766777, −7.69235598353057144113433688931, −6.60290209243787658100563977365, −5.99957414725083822959617513677, −4.64441589000443507488521183376, −3.49871863410090690759635520335, −2.65042841336486185508883232036, −1.56466500627473729904824421260, 0.62646754899912878357522714102, 2.22778549934075934785021598048, 3.42652244715687296518268802658, 4.05367375632900908767603345882, 5.12638041318909674563462777712, 6.65135504461592436899533542821, 7.14832027640051293736749677729, 8.076809740645782658336184260258, 8.974516772037982661897743281442, 9.766345752428768138937674198252

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