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Properties

Label 2-15e2-5.4-c7-0-24
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  + 15.7i·2-s − 120.·4-s + 34.4i·7-s + 121. i·8-s + 3.96e3·11-s + 5.60e3i·13-s − 542.·14-s − 1.73e4·16-s − 1.99e4i·17-s + 4.99e4·19-s + 6.24e4i·22-s − 1.09e5i·23-s − 8.83e4·26-s − 4.14e3i·28-s + 1.92e5·29-s + ⋯
L(s)  = 1  + 1.39i·2-s − 0.939·4-s + 0.0379i·7-s + 0.0837i·8-s + 0.897·11-s + 0.707i·13-s − 0.0528·14-s − 1.05·16-s − 0.982i·17-s + 1.67·19-s + 1.25i·22-s − 1.88i·23-s − 0.985·26-s − 0.0356i·28-s + 1.46·29-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

\(L(4)\) \(\approx\) \(2.420812590\)
\(L(\frac12)\) \(\approx\) \(2.420812590\)
\(L(\frac{9}{2})\) 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 \( 1 - 15.7iT - 128T^{2} \)
7 \( 1 - 34.4iT - 8.23e5T^{2} \)
11 \( 1 - 3.96e3T + 1.94e7T^{2} \)
13 \( 1 - 5.60e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.99e4iT - 4.10e8T^{2} \)
19 \( 1 - 4.99e4T + 8.93e8T^{2} \)
23 \( 1 + 1.09e5iT - 3.40e9T^{2} \)
29 \( 1 - 1.92e5T + 1.72e10T^{2} \)
31 \( 1 - 1.25e5T + 2.75e10T^{2} \)
37 \( 1 - 7.43e4iT - 9.49e10T^{2} \)
41 \( 1 + 5.77e5T + 1.94e11T^{2} \)
43 \( 1 - 2.64e5iT - 2.71e11T^{2} \)
47 \( 1 + 3.06e5iT - 5.06e11T^{2} \)
53 \( 1 - 4.46e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.97e6T + 2.48e12T^{2} \)
61 \( 1 + 1.27e6T + 3.14e12T^{2} \)
67 \( 1 - 4.12e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.81e6T + 9.09e12T^{2} \)
73 \( 1 - 4.01e6iT - 1.10e13T^{2} \)
79 \( 1 - 1.32e6T + 1.92e13T^{2} \)
83 \( 1 - 1.91e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.00e6T + 4.42e13T^{2} \)
97 \( 1 - 3.89e6iT - 8.07e13T^{2} \)
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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

−11.51355541114629267104353111233, −10.03166839593843714442859498396, −9.013632992201285209792132222865, −8.229654861455619193627158250894, −6.99614588616581928351046795599, −6.54806219016323654647546534355, −5.27366995458270352275682031688, −4.35357191689740150340510669542, −2.66917995629102675971266168448, −0.898396346144851675230545645353, 0.814973160406954973559761704327, 1.64654424709276606288962593271, 3.09317842944145676434825262458, 3.81658100018673203283756549064, 5.22156326652218239123584645169, 6.57906358886868567681739730830, 7.84498998690371004183621979472, 9.120744411135915329430567602760, 9.916959798427501802322011636748, 10.70406574393632592795422038183

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