↑

Properties

Label 2-15e2-5.4-c7-0-21
Degree $2$
Conductor $225$
Sign $0.894 + 0.447i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 7.13i·2-s + 77.0·4-s + 112. i·7-s − 1.46e3i·8-s − 656.·11-s + 7.80e3i·13-s + 805.·14-s − 575.·16-s + 1.46e4i·17-s + 4.25e4·19-s + 4.68e3i·22-s − 5.31e4i·23-s + 5.56e4·26-s + 8.70e3i·28-s − 1.67e5·29-s + ⋯
L(s)  = 1  − 0.630i·2-s + 0.602·4-s + 0.124i·7-s − 1.01i·8-s − 0.148·11-s + 0.985i·13-s + 0.0784·14-s − 0.0351·16-s + 0.722i·17-s + 1.42·19-s + 0.0937i·22-s − 0.910i·23-s + 0.621·26-s + 0.0749i·28-s − 1.27·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(70.2866\)
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: \((2,\ 225,\ (\ :7/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.734054759\)
\(L(\frac12)\) \(\approx\) \(2.734054759\)
\(L(\frac{9}{2})\) 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 \( 1 + 7.13iT - 128T^{2} \)
7 \( 1 - 112. iT - 8.23e5T^{2} \)
11 \( 1 + 656.T + 1.94e7T^{2} \)
13 \( 1 - 7.80e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.46e4iT - 4.10e8T^{2} \)
19 \( 1 - 4.25e4T + 8.93e8T^{2} \)
23 \( 1 + 5.31e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.67e5T + 1.72e10T^{2} \)
31 \( 1 - 1.08e5T + 2.75e10T^{2} \)
37 \( 1 - 4.24e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.34e5T + 1.94e11T^{2} \)
43 \( 1 - 6.44e5iT - 2.71e11T^{2} \)
47 \( 1 + 4.00e5iT - 5.06e11T^{2} \)
53 \( 1 - 9.34e5iT - 1.17e12T^{2} \)
59 \( 1 + 8.42e5T + 2.48e12T^{2} \)
61 \( 1 - 2.67e6T + 3.14e12T^{2} \)
67 \( 1 + 2.22e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.04e5T + 9.09e12T^{2} \)
73 \( 1 + 1.39e6iT - 1.10e13T^{2} \)
79 \( 1 + 2.77e6T + 1.92e13T^{2} \)
83 \( 1 - 7.48e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.28e7T + 4.42e13T^{2} \)
97 \( 1 + 2.96e6iT - 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.05519003642532828969317225705, −10.05971861039721578761488960298, −9.209543514505889494165842807324, −7.87832620287716774545517383802, −6.85926421634696060856665093354, −5.87998468446260864477752487576, −4.38036417760117341576378619760, −3.18649960416655036298746314304, −2.07579883552351891157777892608, −0.968257498990009881973467604268, 0.790692045962651764615952356740, 2.32212526378494639609993252803, 3.48566962485396724102059713196, 5.23293624194907815014597858507, 5.85974693856740997740802301365, 7.33946993485787353406830032825, 7.61693516328444487061104469451, 8.992548826134875126327557496963, 10.09517826535112363633112322739, 11.13997173977422042352758178487

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