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Properties

Label 2-15e2-15.14-c8-0-5
Degree $2$
Conductor $225$
Sign $-0.988 - 0.151i$
Analytic cond. $91.6601$
Root an. cond. $9.57393$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.3·2-s + 385.·4-s + 2.90e3i·7-s + 3.29e3·8-s − 6.32e3i·11-s + 7.67e3i·13-s + 7.36e4i·14-s − 1.54e4·16-s − 1.55e5·17-s − 3.85e4·19-s − 1.60e5i·22-s − 2.26e5·23-s + 1.94e5i·26-s + 1.12e6i·28-s − 3.82e5i·29-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.50·4-s + 1.21i·7-s + 0.803·8-s − 0.432i·11-s + 0.268i·13-s + 1.91i·14-s − 0.235·16-s − 1.85·17-s − 0.295·19-s − 0.684i·22-s − 0.810·23-s + 0.425i·26-s + 1.82i·28-s − 0.540i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.988 - 0.151i$
Analytic conductor: \(91.6601\)
Root analytic conductor: \(9.57393\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :4),\ -0.988 - 0.151i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.298543030\)
\(L(\frac12)\) \(\approx\) \(1.298543030\)
\(L(5)\) 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 \( 1 - 25.3T + 256T^{2} \)
7 \( 1 - 2.90e3iT - 5.76e6T^{2} \)
11 \( 1 + 6.32e3iT - 2.14e8T^{2} \)
13 \( 1 - 7.67e3iT - 8.15e8T^{2} \)
17 \( 1 + 1.55e5T + 6.97e9T^{2} \)
19 \( 1 + 3.85e4T + 1.69e10T^{2} \)
23 \( 1 + 2.26e5T + 7.83e10T^{2} \)
29 \( 1 + 3.82e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.30e6T + 8.52e11T^{2} \)
37 \( 1 - 2.63e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.01e6iT - 7.98e12T^{2} \)
43 \( 1 + 6.01e6iT - 1.16e13T^{2} \)
47 \( 1 + 7.71e6T + 2.38e13T^{2} \)
53 \( 1 + 6.38e6T + 6.22e13T^{2} \)
59 \( 1 - 2.20e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.90e7T + 1.91e14T^{2} \)
67 \( 1 - 2.29e6iT - 4.06e14T^{2} \)
71 \( 1 - 2.97e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.03e7iT - 8.06e14T^{2} \)
79 \( 1 - 1.24e7T + 1.51e15T^{2} \)
83 \( 1 - 2.06e7T + 2.25e15T^{2} \)
89 \( 1 - 4.98e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.30e8iT - 7.83e15T^{2} \)
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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

−11.78968598507652854934942112768, −10.68543944243861084714574684024, −9.196584150592179714931904924383, −8.326907105005283805569155912782, −6.64956117298058444789666876998, −6.06776672682575886529403708846, −4.99180319173381626352711131763, −4.11068033978342015083263354419, −2.81093323327755993882929982555, −2.02892845321525657082550377504, 0.13745512805977168915652375445, 1.84062935475770349910426797888, 3.09576471472186289537103107621, 4.28030634265883343641648158829, 4.71492889899601886732204012408, 6.22799816442700499771144527268, 6.86500750946042122893469637200, 8.055876059904330999998239957221, 9.549112459037896683877104202280, 10.75375827967297981865770591854

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