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Properties

Label 2-15e2-5.4-c5-0-20
Degree $2$
Conductor $225$
Sign $-0.894 + 0.447i$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
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  − 10.2i·2-s − 73.3·4-s + 68.9i·7-s + 423. i·8-s + 486.·11-s + 428. i·13-s + 707.·14-s + 2.00e3·16-s − 1.80e3i·17-s + 1.04e3·19-s − 4.98e3i·22-s + 686. i·23-s + 4.39e3·26-s − 5.05e3i·28-s − 1.33e3·29-s + ⋯
L(s)  = 1  − 1.81i·2-s − 2.29·4-s + 0.531i·7-s + 2.34i·8-s + 1.21·11-s + 0.703i·13-s + 0.964·14-s + 1.95·16-s − 1.51i·17-s + 0.665·19-s − 2.19i·22-s + 0.270i·23-s + 1.27·26-s − 1.21i·28-s − 0.295·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.717348089\)
\(L(\frac12)\) \(\approx\) \(1.717348089\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 10.2iT - 32T^{2} \)
7 \( 1 - 68.9iT - 1.68e4T^{2} \)
11 \( 1 - 486.T + 1.61e5T^{2} \)
13 \( 1 - 428. iT - 3.71e5T^{2} \)
17 \( 1 + 1.80e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.04e3T + 2.47e6T^{2} \)
23 \( 1 - 686. iT - 6.43e6T^{2} \)
29 \( 1 + 1.33e3T + 2.05e7T^{2} \)
31 \( 1 - 7.99e3T + 2.86e7T^{2} \)
37 \( 1 + 1.97e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.07e4T + 1.15e8T^{2} \)
43 \( 1 + 1.50e4iT - 1.47e8T^{2} \)
47 \( 1 + 895. iT - 2.29e8T^{2} \)
53 \( 1 + 1.93e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.11e4T + 7.14e8T^{2} \)
61 \( 1 + 2.77e4T + 8.44e8T^{2} \)
67 \( 1 + 7.71e3iT - 1.35e9T^{2} \)
71 \( 1 - 5.14e4T + 1.80e9T^{2} \)
73 \( 1 - 4.37e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.22e3T + 3.07e9T^{2} \)
83 \( 1 - 5.29e4iT - 3.93e9T^{2} \)
89 \( 1 - 4.46e4T + 5.58e9T^{2} \)
97 \( 1 + 1.48e5iT - 8.58e9T^{2} \)
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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

−11.34861770490595450009812345013, −9.967536358593570255243689109281, −9.358285209076702835204477840321, −8.586516972408714862982526622091, −6.90763930249913597638688663613, −5.27321099110046361834509340222, −4.16963355513791324110101473087, −3.05989958277197480007175047168, −1.89730861216422122476503051963, −0.68530553843649821238219290537, 0.997806139958791108548492616526, 3.67656684231702198026530066436, 4.69471675327669928840477821988, 5.97418006161981908271677073890, 6.64034694069303821836917608690, 7.72440938581944937335599353887, 8.472570128531312430481978955318, 9.491718332578534921140869202322, 10.50989709691709998841117394910, 12.01825704890352928588207178313

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