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Properties

Label 2-30e2-9.5-c2-0-16
Degree $2$
Conductor $900$
Sign $-0.128 - 0.991i$
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  + (−0.841 + 2.87i)3-s + (5.98 + 10.3i)7-s + (−7.58 − 4.84i)9-s + (7.43 − 4.29i)11-s + (8.35 − 14.4i)13-s + 22.1i·17-s + 26.9·19-s + (−34.9 + 8.51i)21-s + (33.6 + 19.3i)23-s + (20.3 − 17.7i)27-s + (8.44 − 4.87i)29-s + (6.01 − 10.4i)31-s + (6.10 + 25.0i)33-s − 25.0·37-s + (34.6 + 36.2i)39-s + ⋯
L(s)  = 1  + (−0.280 + 0.959i)3-s + (0.855 + 1.48i)7-s + (−0.842 − 0.538i)9-s + (0.675 − 0.390i)11-s + (0.642 − 1.11i)13-s + 1.30i·17-s + 1.41·19-s + (−1.66 + 0.405i)21-s + (1.46 + 0.843i)23-s + (0.753 − 0.657i)27-s + (0.291 − 0.168i)29-s + (0.194 − 0.336i)31-s + (0.184 + 0.758i)33-s − 0.677·37-s + (0.887 + 0.928i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.128 - 0.991i$
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: \((2,\ 900,\ (\ :1),\ -0.128 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.062578753\)
\(L(\frac12)\) \(\approx\) \(2.062578753\)
\(L(2)\) 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 \( 1 + (0.841 - 2.87i)T \)
5 \( 1 \)
good7 \( 1 + (-5.98 - 10.3i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-7.43 + 4.29i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.35 + 14.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 22.1iT - 289T^{2} \)
19 \( 1 - 26.9T + 361T^{2} \)
23 \( 1 + (-33.6 - 19.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-8.44 + 4.87i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.01 + 10.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 25.0T + 1.36e3T^{2} \)
41 \( 1 + (-34.8 - 20.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (39.1 + 67.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (64.3 - 37.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 13.0iT - 2.80e3T^{2} \)
59 \( 1 + (-32.9 - 18.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (4.56 + 7.90i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-12.0 + 20.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 43.7iT - 5.04e3T^{2} \)
73 \( 1 - 7.68T + 5.32e3T^{2} \)
79 \( 1 + (-19.3 - 33.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (73.0 - 42.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 61.5iT - 7.92e3T^{2} \)
97 \( 1 + (-31.3 - 54.2i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.13691345394851198031141424068, −9.209325834506819920916580690456, −8.625019899323045130190328529839, −7.932840697034046077253627443127, −6.38509510822134935757299754566, −5.50629300566296891353591206662, −5.15204582087389053353348959648, −3.74396555052997493515692698996, −2.92196146156226122414084621865, −1.28571495694251888313615972498, 0.857847872891334090947170684651, 1.56902482812828934153358955192, 3.15114572168266904686022783754, 4.47640047949453791647798801984, 5.15669258066911482737992926577, 6.73053499393556486223080470020, 6.94781923254983065553501343726, 7.77625737585630225250806115048, 8.743587866391417680420571633633, 9.656066932921010829370555980359

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