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Properties

Label 2-30e2-9.5-c2-0-29
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: $\chi_{900} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.973 + 0.227i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.140752163\)
\(L(\frac12)\) \(\approx\) \(2.140752163\)
\(L(2)\) 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

−9.961731262150930416289245405468, −8.870230224639191733822860102957, −8.574645472840571081993442995562, −7.49392236257507108540215722371, −6.14184109489615895874159055411, −5.60315814133742484080751511142, −4.28786774963581074593424058094, −3.61659252100697835242940380305, −2.56505531890622571127080882807, −0.73523327283136070728652773206, 1.35046072904087825829941593988, 2.00270784725072794232489227623, 3.65174322724690705931894901183, 4.33128967437898420184419030782, 6.01887981846940841920406021177, 6.50483000552765781647419767983, 7.32467126002858798750151505656, 8.258617838435479738203567031734, 9.075827890754187021153548012880, 9.664789636058963161122842680598

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