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Properties

Label 2-30e2-20.19-c2-0-64
Degree $2$
Conductor $900$
Sign $-0.367 + 0.930i$
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  + (−1.47 − 1.35i)2-s + (0.350 + 3.98i)4-s + 10.9·7-s + (4.86 − 6.35i)8-s − 11.3i·11-s − 10.8i·13-s + (−16.1 − 14.7i)14-s + (−15.7 + 2.79i)16-s + 15.8i·17-s − 24.9i·19-s + (−15.3 + 16.7i)22-s − 20.9·23-s + (−14.5 + 15.9i)26-s + (3.83 + 43.5i)28-s + 22.8·29-s + ⋯
L(s)  = 1  + (−0.737 − 0.675i)2-s + (0.0876 + 0.996i)4-s + 1.55·7-s + (0.608 − 0.793i)8-s − 1.03i·11-s − 0.831i·13-s + (−1.15 − 1.05i)14-s + (−0.984 + 0.174i)16-s + 0.929i·17-s − 1.31i·19-s + (−0.697 + 0.761i)22-s − 0.911·23-s + (−0.561 + 0.613i)26-s + (0.136 + 1.55i)28-s + 0.786·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.367 + 0.930i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.367 + 0.930i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.354274975\)
\(L(\frac12)\) \(\approx\) \(1.354274975\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.47 + 1.35i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 10.9T + 49T^{2} \)
11 \( 1 + 11.3iT - 121T^{2} \)
13 \( 1 + 10.8iT - 169T^{2} \)
17 \( 1 - 15.8iT - 289T^{2} \)
19 \( 1 + 24.9iT - 361T^{2} \)
23 \( 1 + 20.9T + 529T^{2} \)
29 \( 1 - 22.8T + 841T^{2} \)
31 \( 1 + 22.7iT - 961T^{2} \)
37 \( 1 - 19.1iT - 1.36e3T^{2} \)
41 \( 1 + 17T + 1.68e3T^{2} \)
43 \( 1 + 6.51T + 1.84e3T^{2} \)
47 \( 1 - 38.9T + 2.20e3T^{2} \)
53 \( 1 + 13.2iT - 2.80e3T^{2} \)
59 \( 1 + 95.6iT - 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 - 54.1T + 4.48e3T^{2} \)
71 \( 1 + 68.5iT - 5.04e3T^{2} \)
73 \( 1 - 44.1iT - 5.32e3T^{2} \)
79 \( 1 - 81.7iT - 6.24e3T^{2} \)
83 \( 1 + 27.9T + 6.88e3T^{2} \)
89 \( 1 + 42.1T + 7.92e3T^{2} \)
97 \( 1 + 154. iT - 9.40e3T^{2} \)
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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

−9.741751753324419789553949122104, −8.546734484585956450067986977298, −8.301483035790517199286074929417, −7.50921536557744033476125147775, −6.26517613415576106031398170255, −5.08697130919113469365843606817, −4.10064055466925188350435556980, −2.92981685907557912167059970714, −1.78420767407725856057893917574, −0.60610505303260894859361680659, 1.35693778327280884110582508218, 2.17389209982257091212655381492, 4.28724731428021525136699031728, 4.94505592893872357412464605994, 5.91715574672106729060914338053, 7.04650566782212249665612762662, 7.65523291517933294700283510112, 8.400877310132580294895969528546, 9.207607505373452249728513003085, 10.09844610035364058501838815306

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