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Properties

Label 2-30e2-20.19-c2-0-63
Degree $2$
Conductor $900$
Sign $-0.987 - 0.156i$
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  + (−0.305 − 1.97i)2-s + (−3.81 + 1.20i)4-s − 0.329·7-s + (3.55 + 7.16i)8-s + 20.4i·11-s − 0.416i·13-s + (0.100 + 0.652i)14-s + (13.0 − 9.21i)16-s − 18.5i·17-s − 12.4i·19-s + (40.5 − 6.26i)22-s − 23.2·23-s + (−0.823 + 0.127i)26-s + (1.25 − 0.398i)28-s − 23.9·29-s + ⋯
L(s)  = 1  + (−0.152 − 0.988i)2-s + (−0.953 + 0.302i)4-s − 0.0471·7-s + (0.444 + 0.895i)8-s + 1.86i·11-s − 0.0320i·13-s + (0.00720 + 0.0465i)14-s + (0.817 − 0.575i)16-s − 1.09i·17-s − 0.655i·19-s + (1.84 − 0.284i)22-s − 1.01·23-s + (−0.0316 + 0.00489i)26-s + (0.0449 − 0.0142i)28-s − 0.824·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.987 - 0.156i$
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: \((2,\ 900,\ (\ :1),\ -0.987 - 0.156i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5204706756\)
\(L(\frac12)\) \(\approx\) \(0.5204706756\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.305 + 1.97i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.329T + 49T^{2} \)
11 \( 1 - 20.4iT - 121T^{2} \)
13 \( 1 + 0.416iT - 169T^{2} \)
17 \( 1 + 18.5iT - 289T^{2} \)
19 \( 1 + 12.4iT - 361T^{2} \)
23 \( 1 + 23.2T + 529T^{2} \)
29 \( 1 + 23.9T + 841T^{2} \)
31 \( 1 + 42.0iT - 961T^{2} \)
37 \( 1 + 50.9iT - 1.36e3T^{2} \)
41 \( 1 + 46.7T + 1.68e3T^{2} \)
43 \( 1 - 55.5T + 1.84e3T^{2} \)
47 \( 1 - 81.7T + 2.20e3T^{2} \)
53 \( 1 - 29.9iT - 2.80e3T^{2} \)
59 \( 1 - 24.3iT - 3.48e3T^{2} \)
61 \( 1 + 74.8T + 3.72e3T^{2} \)
67 \( 1 + 72.8T + 4.48e3T^{2} \)
71 \( 1 + 39.2iT - 5.04e3T^{2} \)
73 \( 1 + 46.5iT - 5.32e3T^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 + 5.88T + 6.88e3T^{2} \)
89 \( 1 + 61.0T + 7.92e3T^{2} \)
97 \( 1 + 95.5iT - 9.40e3T^{2} \)
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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

−9.473935820650006041488939249731, −9.107368607656929162270059670074, −7.68905853845908876530517629237, −7.27374383934988380500169230610, −5.77115906218370379893223643610, −4.68789857317711261874206686265, −4.05785699472629833283320502054, −2.65764882528249396578456845859, −1.83387927749610307783024575642, −0.18781047299011237784988792107, 1.32867838817874305884086653546, 3.30037389189440508937186390986, 4.13837639572545637377359320985, 5.49045206143374253194047313117, 6.00169051438638544150293798650, 6.83827684119246753161827087831, 8.090470209393602243504405887136, 8.369169216610544896566474365767, 9.283166008291043471220721397699, 10.29730717016636931324588609927

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