Properties

Label 2-30e2-9.2-c2-0-13
Degree $2$
Conductor $900$
Sign $0.941 - 0.337i$
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  + (−2.89 − 0.783i)3-s + (2.35 − 4.08i)7-s + (7.77 + 4.53i)9-s + (16.6 + 9.63i)11-s + (−4.68 − 8.11i)13-s + 16.2i·17-s − 19.0·19-s + (−10.0 + 9.97i)21-s + (−14.3 + 8.30i)23-s + (−18.9 − 19.2i)27-s + (−3.87 − 2.23i)29-s + (−7.19 − 12.4i)31-s + (−40.7 − 40.9i)33-s + 55.6·37-s + (7.21 + 27.1i)39-s + ⋯
L(s)  = 1  + (−0.965 − 0.261i)3-s + (0.336 − 0.583i)7-s + (0.863 + 0.504i)9-s + (1.51 + 0.875i)11-s + (−0.360 − 0.624i)13-s + 0.955i·17-s − 1.00·19-s + (−0.477 + 0.474i)21-s + (−0.625 + 0.361i)23-s + (−0.702 − 0.712i)27-s + (−0.133 − 0.0771i)29-s + (−0.232 − 0.401i)31-s + (−1.23 − 1.24i)33-s + 1.50·37-s + (0.184 + 0.696i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.941 - 0.337i$
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: \((2,\ 900,\ (\ :1),\ 0.941 - 0.337i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.378549224\)
\(L(\frac12)\) \(\approx\) \(1.378549224\)
\(L(2)\) 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 \( 1 + (2.89 + 0.783i)T \)
5 \( 1 \)
good7 \( 1 + (-2.35 + 4.08i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-16.6 - 9.63i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4.68 + 8.11i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 16.2iT - 289T^{2} \)
19 \( 1 + 19.0T + 361T^{2} \)
23 \( 1 + (14.3 - 8.30i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.87 + 2.23i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (7.19 + 12.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 55.6T + 1.36e3T^{2} \)
41 \( 1 + (-26.7 + 15.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (21.5 - 37.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-35.9 - 20.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 30.7iT - 2.80e3T^{2} \)
59 \( 1 + (-74.4 + 42.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (51.7 - 89.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-61.7 - 107. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 111. iT - 5.04e3T^{2} \)
73 \( 1 - 90.5T + 5.32e3T^{2} \)
79 \( 1 + (-41.6 + 72.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (31.0 + 17.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 84.9iT - 7.92e3T^{2} \)
97 \( 1 + (52.6 - 91.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.08997758928284535195094555037, −9.312947391232901374533007780056, −8.043635227613505919938705335938, −7.33563467814036001091319456433, −6.44641580261524826425235270758, −5.78451624162627955183510678230, −4.45452132190577529106016178354, −4.02829047883255958719980014686, −2.04854420156833279352142334515, −0.978444118735366540009231876771, 0.67073357700267167684823409713, 2.09622234899349444037734120589, 3.73467176520999964220148260920, 4.55326031730789099237215474427, 5.55493066286171685835751998021, 6.38136100748556432623289894905, 6.99756154589835198895548065603, 8.343196295024637501813784170860, 9.161605884648978815632923854006, 9.794850604875108800162761198788

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