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Properties

Label 2-30e2-25.16-c1-0-3
Degree $2$
Conductor $900$
Sign $0.177 - 0.984i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.96 + 1.06i)5-s − 1.74·7-s + (1.87 + 1.36i)11-s + (−3.99 + 2.90i)13-s + (−1.15 + 3.55i)17-s + (−0.523 + 1.61i)19-s + (7.02 + 5.10i)23-s + (2.72 + 4.19i)25-s + (−0.964 − 2.96i)29-s + (2.95 − 9.09i)31-s + (−3.41 − 1.85i)35-s + (−4.34 + 3.15i)37-s + (−7.05 + 5.12i)41-s + 2.86·43-s + (2.61 + 8.04i)47-s + ⋯
L(s)  = 1  + (0.878 + 0.477i)5-s − 0.657·7-s + (0.565 + 0.411i)11-s + (−1.10 + 0.805i)13-s + (−0.280 + 0.862i)17-s + (−0.120 + 0.369i)19-s + (1.46 + 1.06i)23-s + (0.544 + 0.838i)25-s + (−0.179 − 0.551i)29-s + (0.530 − 1.63i)31-s + (−0.578 − 0.314i)35-s + (−0.713 + 0.518i)37-s + (−1.10 + 0.800i)41-s + 0.436·43-s + (0.381 + 1.17i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.177 - 0.984i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.177 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15651 + 0.966555i\)
\(L(\frac12)\) \(\approx\) \(1.15651 + 0.966555i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.96 - 1.06i)T \)
good7 \( 1 + 1.74T + 7T^{2} \)
11 \( 1 + (-1.87 - 1.36i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.99 - 2.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.15 - 3.55i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.523 - 1.61i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-7.02 - 5.10i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.964 + 2.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.95 + 9.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.34 - 3.15i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.05 - 5.12i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.86T + 43T^{2} \)
47 \( 1 + (-2.61 - 8.04i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.415 - 1.27i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.54 + 2.57i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-12.4 - 9.01i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.85 + 8.77i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.00728 + 0.0224i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.827 - 0.601i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.246 + 0.759i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.732 - 2.25i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (3.93 + 2.85i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.06 + 3.29i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.943646434301514772780437228739, −9.698647053122217225626595886120, −8.872282997166379151067464505462, −7.57326557360933483625056709546, −6.75127655099958889107441336943, −6.16320380190828142246137981077, −5.07175403458901702630400187149, −3.94880335235016970814589864420, −2.75257374600813016091534738250, −1.68717985725575760088026422526, 0.71978409154878265357233812524, 2.38047667598763944594676362608, 3.32417272050111545433947884618, 4.89328825047872996284808048972, 5.35071724281002388057279001272, 6.66192386396333273147355992917, 7.05736300303392149202156358421, 8.621110223284543990836809406541, 8.985478300238601396883871089876, 9.973659376596488090087693334623

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