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Properties

Label 2-15e2-5.4-c7-0-6
Degree $2$
Conductor $225$
Sign $-0.447 + 0.894i$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.75i·2-s + 51.3·4-s + 1.33e3i·7-s + 1.57e3i·8-s − 7.41e3·11-s + 1.45e4i·13-s − 1.17e4·14-s − 7.18e3·16-s − 1.44e4i·17-s − 409.·19-s − 6.49e4i·22-s − 1.79e4i·23-s − 1.27e5·26-s + 6.86e4i·28-s − 1.07e4·29-s + ⋯
L(s)  = 1  + 0.774i·2-s + 0.400·4-s + 1.47i·7-s + 1.08i·8-s − 1.67·11-s + 1.84i·13-s − 1.14·14-s − 0.438·16-s − 0.711i·17-s − 0.0137·19-s − 1.29i·22-s − 0.307i·23-s − 1.42·26-s + 0.591i·28-s − 0.0815·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.065670729\)
\(L(\frac12)\) \(\approx\) \(1.065670729\)
\(L(\frac{9}{2})\) 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 \( 1 - 8.75iT - 128T^{2} \)
7 \( 1 - 1.33e3iT - 8.23e5T^{2} \)
11 \( 1 + 7.41e3T + 1.94e7T^{2} \)
13 \( 1 - 1.45e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.44e4iT - 4.10e8T^{2} \)
19 \( 1 + 409.T + 8.93e8T^{2} \)
23 \( 1 + 1.79e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.07e4T + 1.72e10T^{2} \)
31 \( 1 - 1.78e5T + 2.75e10T^{2} \)
37 \( 1 + 4.27e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.33e4T + 1.94e11T^{2} \)
43 \( 1 + 8.93e4iT - 2.71e11T^{2} \)
47 \( 1 - 1.61e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.21e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.69e6T + 2.48e12T^{2} \)
61 \( 1 + 1.23e6T + 3.14e12T^{2} \)
67 \( 1 + 9.44e5iT - 6.06e12T^{2} \)
71 \( 1 - 9.36e5T + 9.09e12T^{2} \)
73 \( 1 - 5.49e6iT - 1.10e13T^{2} \)
79 \( 1 - 3.29e6T + 1.92e13T^{2} \)
83 \( 1 + 4.16e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.50e6T + 4.42e13T^{2} \)
97 \( 1 - 6.61e6iT - 8.07e13T^{2} \)
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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

−11.69318686992216577693462834004, −10.80556042939849341042943494115, −9.421099476462032501979638584252, −8.559947019385085274002827207191, −7.60661025716771476418472252451, −6.56650846157775876800533199566, −5.63155244330702626865489479878, −4.77350819540683774112234239078, −2.70961632303359204266902207124, −2.05904018440763886508813155297, 0.24339537016772384489159020424, 1.19883231530534332634185322028, 2.71233906485702133574108978818, 3.52962007432623672677538825839, 4.91423390328074320023416058288, 6.25025466440418532718363023644, 7.55093596136980897416508569557, 8.040221825921616693381530738099, 10.00443658638937432362250879599, 10.44083112022655583731703357632

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