Properties

Label 2-30e2-5.4-c5-0-19
Degree $2$
Conductor $900$
Sign $0.447 + 0.894i$
Analytic cond. $144.345$
Root an. cond. $12.0143$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 88i·7-s − 540·11-s − 418i·13-s + 594i·17-s − 836·19-s + 4.10e3i·23-s − 594·29-s + 4.25e3·31-s + 298i·37-s − 1.72e4·41-s − 1.21e4i·43-s − 1.29e3i·47-s + 9.06e3·49-s − 1.94e4i·53-s − 7.66e3·59-s + ⋯
L(s)  = 1  + 0.678i·7-s − 1.34·11-s − 0.685i·13-s + 0.498i·17-s − 0.531·19-s + 1.61i·23-s − 0.131·29-s + 0.795·31-s + 0.0357i·37-s − 1.60·41-s − 0.997i·43-s − 0.0855i·47-s + 0.539·49-s − 0.953i·53-s − 0.286·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(144.345\)
Root analytic conductor: \(12.0143\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :5/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.031853890\)
\(L(\frac12)\) \(\approx\) \(1.031853890\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good7 \( 1 - 88iT - 1.68e4T^{2} \)
11 \( 1 + 540T + 1.61e5T^{2} \)
13 \( 1 + 418iT - 3.71e5T^{2} \)
17 \( 1 - 594iT - 1.41e6T^{2} \)
19 \( 1 + 836T + 2.47e6T^{2} \)
23 \( 1 - 4.10e3iT - 6.43e6T^{2} \)
29 \( 1 + 594T + 2.05e7T^{2} \)
31 \( 1 - 4.25e3T + 2.86e7T^{2} \)
37 \( 1 - 298iT - 6.93e7T^{2} \)
41 \( 1 + 1.72e4T + 1.15e8T^{2} \)
43 \( 1 + 1.21e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.29e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.94e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.66e3T + 7.14e8T^{2} \)
61 \( 1 + 3.47e4T + 8.44e8T^{2} \)
67 \( 1 + 2.18e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.68e4T + 1.80e9T^{2} \)
73 \( 1 - 6.75e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.69e4T + 3.07e9T^{2} \)
83 \( 1 + 6.77e4iT - 3.93e9T^{2} \)
89 \( 1 - 2.97e4T + 5.58e9T^{2} \)
97 \( 1 - 1.22e5iT - 8.58e9T^{2} \)
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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

−9.196774681840523351866965026866, −8.259185652735669874548909017127, −7.74288631113524596028535444964, −6.60395728824762022453159269160, −5.55772141231612720390430138622, −5.09103315033385085129811913772, −3.69213032833222609434718102039, −2.73957655313444446460638721807, −1.75698941245322813925903634857, −0.26587996075051842240415559431, 0.73163819836755203867357653958, 2.12426163975535744952458126415, 3.05855191116269891629839567893, 4.34674078828028467722374994307, 4.94972971909911513305244724992, 6.18769132308770942256721161814, 6.96836180139172492263137897976, 7.86055489867944440276876455033, 8.574881558706222273816302232123, 9.599947340749524307770283161625

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