↑

Properties

Label 2-30e2-4.3-c2-0-19
Degree $2$
Conductor $900$
Sign $-0.548 - 0.836i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.950 + 1.75i)2-s + (−2.19 − 3.34i)4-s − 3.98i·7-s + (7.97 − 0.677i)8-s + 8.39i·11-s − 9.77·13-s + (7.02 + 3.79i)14-s + (−6.38 + 14.6i)16-s − 12.1·17-s − 17.3i·19-s + (−14.7 − 7.97i)22-s − 2.97i·23-s + (9.28 − 17.1i)26-s + (−13.3 + 8.74i)28-s + 51.6·29-s + ⋯
L(s)  = 1  + (−0.475 + 0.879i)2-s + (−0.548 − 0.836i)4-s − 0.569i·7-s + (0.996 − 0.0847i)8-s + 0.763i·11-s − 0.751·13-s + (0.501 + 0.270i)14-s + (−0.399 + 0.916i)16-s − 0.714·17-s − 0.914i·19-s + (−0.671 − 0.362i)22-s − 0.129i·23-s + (0.357 − 0.661i)26-s + (−0.476 + 0.312i)28-s + 1.78·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.548 - 0.836i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.548 - 0.836i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9184761033\)
\(L(\frac12)\) \(\approx\) \(0.9184761033\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.950 - 1.75i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 3.98iT - 49T^{2} \)
11 \( 1 - 8.39iT - 121T^{2} \)
13 \( 1 + 9.77T + 169T^{2} \)
17 \( 1 + 12.1T + 289T^{2} \)
19 \( 1 + 17.3iT - 361T^{2} \)
23 \( 1 + 2.97iT - 529T^{2} \)
29 \( 1 - 51.6T + 841T^{2} \)
31 \( 1 - 30.7iT - 961T^{2} \)
37 \( 1 + 19.5T + 1.36e3T^{2} \)
41 \( 1 - 42.5T + 1.68e3T^{2} \)
43 \( 1 - 62.9iT - 1.84e3T^{2} \)
47 \( 1 - 39.5iT - 2.20e3T^{2} \)
53 \( 1 - 21.2T + 2.80e3T^{2} \)
59 \( 1 - 50.3iT - 3.48e3T^{2} \)
61 \( 1 + 70.3T + 3.72e3T^{2} \)
67 \( 1 - 94.8iT - 4.48e3T^{2} \)
71 \( 1 - 132. iT - 5.04e3T^{2} \)
73 \( 1 + 77.7T + 5.32e3T^{2} \)
79 \( 1 + 48.3iT - 6.24e3T^{2} \)
83 \( 1 + 140. iT - 6.88e3T^{2} \)
89 \( 1 + 18.1T + 7.92e3T^{2} \)
97 \( 1 - 15T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.13844452732174781211474120635, −9.229980141468251339368625471923, −8.536094842891748939709673441936, −7.44246459410591833071140547359, −7.00427352953158722237997030759, −6.11032450260987241950514522322, −4.80691043991502306829183257321, −4.41253682215310005808613099851, −2.61033757453325136012846721343, −1.07026169014598863535451684506, 0.41425600031053480117716474465, 1.96879216654936268189675487155, 2.90472437482998603738831831308, 3.99770727079493599499294238517, 5.04313759109919172360076867323, 6.13304742506167504317915342507, 7.30799033691976774187972368056, 8.243279937727689710694316184215, 8.848004987879443130952678678237, 9.684043923153347406266709027419

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