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Properties

Label 2-30e2-45.14-c2-0-13
Degree $2$
Conductor $900$
Sign $0.988 + 0.149i$
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  + (−0.386 − 2.97i)3-s + (4.03 − 2.32i)7-s + (−8.70 + 2.30i)9-s + (−12.4 + 7.21i)11-s + (21.2 + 12.2i)13-s − 6.56·17-s + 33.0·19-s + (−8.48 − 11.0i)21-s + (−18.4 + 31.9i)23-s + (10.2 + 24.9i)27-s + (17.4 − 10.0i)29-s + (6.48 − 11.2i)31-s + (26.2 + 34.3i)33-s + 19.9i·37-s + (28.3 − 68.0i)39-s + ⋯
L(s)  = 1  + (−0.128 − 0.991i)3-s + (0.575 − 0.332i)7-s + (−0.966 + 0.255i)9-s + (−1.13 + 0.655i)11-s + (1.63 + 0.945i)13-s − 0.386·17-s + 1.74·19-s + (−0.403 − 0.528i)21-s + (−0.802 + 1.38i)23-s + (0.378 + 0.925i)27-s + (0.602 − 0.348i)29-s + (0.209 − 0.362i)31-s + (0.796 + 1.04i)33-s + 0.538i·37-s + (0.726 − 1.74i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.988 + 0.149i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.988 + 0.149i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.809095002\)
\(L(\frac12)\) \(\approx\) \(1.809095002\)
\(L(2)\) 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 \( 1 + (0.386 + 2.97i)T \)
5 \( 1 \)
good7 \( 1 + (-4.03 + 2.32i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (12.4 - 7.21i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-21.2 - 12.2i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 6.56T + 289T^{2} \)
19 \( 1 - 33.0T + 361T^{2} \)
23 \( 1 + (18.4 - 31.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-17.4 + 10.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.48 + 11.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 19.9iT - 1.36e3T^{2} \)
41 \( 1 + (-44.9 - 25.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-6.41 + 3.70i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (13.6 + 23.7i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 79.3T + 2.80e3T^{2} \)
59 \( 1 + (63.3 + 36.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (12.3 + 21.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (35.4 + 20.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 117. iT - 5.04e3T^{2} \)
73 \( 1 - 40.1iT - 5.32e3T^{2} \)
79 \( 1 + (-11.1 - 19.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-67.3 - 116. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 4.69iT - 7.92e3T^{2} \)
97 \( 1 + (-120. + 69.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.885841522866682549801426957478, −8.934238005577109022265926921445, −7.86587794967635552024595337213, −7.58886786304112409528246736919, −6.47749485766604160149680626793, −5.66181929422108214760966185689, −4.66376934968908857441913953932, −3.37231545090601963212620537208, −2.03844609966433666822487028640, −1.08262619175288064641817389821, 0.73939592917190275276035022594, 2.67342906676692646513862430428, 3.51953556998387747342466240856, 4.67476187499222586824089915856, 5.56954944901666429514150558239, 6.06943855230251240556988902986, 7.65821566937516348810986670607, 8.460644542948894733346716036633, 8.925902458989757451031256202802, 10.21521734528943192280906692071

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