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Properties

Label 2-15e2-15.14-c4-0-22
Degree $2$
Conductor $225$
Sign $0.881 + 0.472i$
Analytic cond. $23.2582$
Root an. cond. $4.82267$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.84·2-s + 45.5·4-s − 71.8i·7-s + 231.·8-s − 66.0i·11-s + 143. i·13-s − 563. i·14-s + 1.09e3·16-s + 88.1·17-s − 397.·19-s − 518. i·22-s + 189.·23-s + 1.12e3i·26-s − 3.27e3i·28-s + 649. i·29-s + ⋯
L(s)  = 1  + 1.96·2-s + 2.84·4-s − 1.46i·7-s + 3.62·8-s − 0.545i·11-s + 0.850i·13-s − 2.87i·14-s + 4.25·16-s + 0.304·17-s − 1.10·19-s − 1.07i·22-s + 0.357·23-s + 1.66i·26-s − 4.17i·28-s + 0.772i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(23.2582\)
Root analytic conductor: \(4.82267\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :2),\ 0.881 + 0.472i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(6.822710890\)
\(L(\frac12)\) \(\approx\) \(6.822710890\)
\(L(3)\) 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 \( 1 - 7.84T + 16T^{2} \)
7 \( 1 + 71.8iT - 2.40e3T^{2} \)
11 \( 1 + 66.0iT - 1.46e4T^{2} \)
13 \( 1 - 143. iT - 2.85e4T^{2} \)
17 \( 1 - 88.1T + 8.35e4T^{2} \)
19 \( 1 + 397.T + 1.30e5T^{2} \)
23 \( 1 - 189.T + 2.79e5T^{2} \)
29 \( 1 - 649. iT - 7.07e5T^{2} \)
31 \( 1 - 508.T + 9.23e5T^{2} \)
37 \( 1 - 1.27e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.53e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.27e3iT - 3.41e6T^{2} \)
47 \( 1 + 3.09e3T + 4.87e6T^{2} \)
53 \( 1 - 940.T + 7.89e6T^{2} \)
59 \( 1 + 5.07e3iT - 1.21e7T^{2} \)
61 \( 1 + 625.T + 1.38e7T^{2} \)
67 \( 1 + 2.76e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.89e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.07e3T + 3.89e7T^{2} \)
83 \( 1 + 1.04e4T + 4.74e7T^{2} \)
89 \( 1 + 1.40e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.51e4iT - 8.85e7T^{2} \)
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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

−11.60771077271009542808862937022, −10.98252815854228357698272314889, −10.09079971311852208612105529134, −8.070356944049759014643455092924, −6.92182069708278835255392495665, −6.33685018480311606671692184145, −4.88975308140431107237633247127, −4.11097399816787183155156658406, −3.09826561904233471903674494313, −1.45458049352714090757294881872, 2.04467575544759035371110663749, 2.94572752534353242763038865867, 4.30300390792019450166998936069, 5.41357719266230606451690008229, 6.03508403076579250175432256803, 7.20754893578675719248442439700, 8.454310042117720365075797444920, 10.10734731463349933760924777219, 11.15548856994053010983096043514, 12.11349202785399576887407728779

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