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Properties

Label 2-30e2-45.29-c2-0-16
Degree $2$
Conductor $900$
Sign $0.285 + 0.958i$
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  + (−2.59 − 1.5i)3-s + (−10.1 − 5.87i)7-s + (4.5 + 7.79i)9-s + (13.1 + 7.57i)11-s + (−15.3 + 8.87i)13-s + 15.1·17-s + 11.2·19-s + (17.6 + 30.5i)21-s + (16.8 + 29.2i)23-s − 27i·27-s + (−8.23 − 4.75i)29-s + (−28.1 − 48.6i)31-s + (−22.7 − 39.3i)33-s − 14i·37-s + 53.2·39-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−1.45 − 0.838i)7-s + (0.5 + 0.866i)9-s + (1.19 + 0.688i)11-s + (−1.18 + 0.682i)13-s + 0.891·17-s + 0.592·19-s + (0.838 + 1.45i)21-s + (0.733 + 1.27i)23-s − i·27-s + (−0.284 − 0.164i)29-s + (−0.906 − 1.57i)31-s + (−0.688 − 1.19i)33-s − 0.378i·37-s + 1.36·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.285 + 0.958i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.285 + 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9318212441\)
\(L(\frac12)\) \(\approx\) \(0.9318212441\)
\(L(2)\) 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 \( 1 + (2.59 + 1.5i)T \)
5 \( 1 \)
good7 \( 1 + (10.1 + 5.87i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-13.1 - 7.57i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (15.3 - 8.87i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 15.1T + 289T^{2} \)
19 \( 1 - 11.2T + 361T^{2} \)
23 \( 1 + (-16.8 - 29.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (8.23 + 4.75i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (28.1 + 48.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 14iT - 1.36e3T^{2} \)
41 \( 1 + (22.5 - 12.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-17.3 - 9.99i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-36.1 + 62.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 37.6T + 2.80e3T^{2} \)
59 \( 1 + (-55.1 + 31.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (0.618 - 1.07i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (74.4 - 42.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 22.1iT - 5.04e3T^{2} \)
73 \( 1 + 60.2iT - 5.32e3T^{2} \)
79 \( 1 + (-51.6 + 89.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-45.0 + 78i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 12.0iT - 7.92e3T^{2} \)
97 \( 1 + (-86.3 - 49.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.594424966993672784385455337254, −9.427710641998060296431133911731, −7.39176173661123576771102244460, −7.31789624018091626960790402463, −6.41102699988662772524215747663, −5.51139822108381596785866584512, −4.35946172930652937023996142332, −3.42275294769675509397109598585, −1.81305982315601407449485935726, −0.48587360948178513112908463987, 0.848862675417501290509463010043, 2.90989473658137376593641469928, 3.61986919763359253604136762308, 4.99951986778225645146474448396, 5.73822594841833991811808324168, 6.50094079361337666087587743802, 7.24076268434071951766781524052, 8.769649622741287582067991987368, 9.357399109789974863330218097741, 10.03861600538991956506563142505

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