Properties

Label 2-30e2-9.2-c2-0-22
Degree $2$
Conductor $900$
Sign $0.973 - 0.227i$
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  + (−0.947 + 2.84i)3-s + (−0.333 + 0.578i)7-s + (−7.20 − 5.39i)9-s + (16.8 + 9.73i)11-s + (−10.0 − 17.3i)13-s − 16.3i·17-s − 2.43·19-s + (−1.32 − 1.49i)21-s + (29.7 − 17.1i)23-s + (22.1 − 15.3i)27-s + (−4.96 − 2.86i)29-s + (13.4 + 23.2i)31-s + (−43.6 + 38.7i)33-s − 11.3·37-s + (58.9 − 12.0i)39-s + ⋯
L(s)  = 1  + (−0.315 + 0.948i)3-s + (−0.0477 + 0.0826i)7-s + (−0.800 − 0.599i)9-s + (1.53 + 0.884i)11-s + (−0.771 − 1.33i)13-s − 0.960i·17-s − 0.128·19-s + (−0.0633 − 0.0713i)21-s + (1.29 − 0.747i)23-s + (0.821 − 0.570i)27-s + (−0.171 − 0.0989i)29-s + (0.433 + 0.750i)31-s + (−1.32 + 1.17i)33-s − 0.307·37-s + (1.51 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.973 - 0.227i$
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: \((2,\ 900,\ (\ :1),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.640533227\)
\(L(\frac12)\) \(\approx\) \(1.640533227\)
\(L(2)\) 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 \( 1 + (0.947 - 2.84i)T \)
5 \( 1 \)
good7 \( 1 + (0.333 - 0.578i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-16.8 - 9.73i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (10.0 + 17.3i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 16.3iT - 289T^{2} \)
19 \( 1 + 2.43T + 361T^{2} \)
23 \( 1 + (-29.7 + 17.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (4.96 + 2.86i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-13.4 - 23.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 11.3T + 1.36e3T^{2} \)
41 \( 1 + (-25.3 + 14.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-17.0 + 29.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (58.8 + 33.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 37.0iT - 2.80e3T^{2} \)
59 \( 1 + (-16.4 + 9.51i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-56.7 + 98.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-53.3 - 92.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 37.0iT - 5.04e3T^{2} \)
73 \( 1 - 115.T + 5.32e3T^{2} \)
79 \( 1 + (-8.38 + 14.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (78.3 + 45.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 28.0iT - 7.92e3T^{2} \)
97 \( 1 + (-4.71 + 8.16i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.880567791367404720203178394913, −9.301837081623455316180772174564, −8.513149921731573102981541017449, −7.24378882091154991292985619838, −6.52008432627382234168369379527, −5.31333572213357660254988727013, −4.71443463714692910588060836915, −3.65799503885041995828262913275, −2.60478137335151982510706881811, −0.72786368427842635834094750806, 0.995240671704410626514642762193, 2.01698902613969940810959732538, 3.43902430199241733108172497607, 4.54743666034044863844476190954, 5.77921286688013763667600576402, 6.56115358769944099998552976663, 7.09925327583520200537319574529, 8.196270523665266300032442243296, 8.982413685631620057899340275916, 9.722588422851133661316955364986

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