↑

Properties

Label 2-15e2-9.7-c1-0-0
Degree $2$
Conductor $225$
Sign $-0.912 - 0.409i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.816 + 1.41i)2-s + (−1.06 − 1.36i)3-s + (−0.334 − 0.579i)4-s + (2.80 − 0.389i)6-s + (−0.252 + 0.437i)7-s − 2.17·8-s + (−0.735 + 2.90i)9-s + (−1.55 + 2.68i)11-s + (−0.435 + 1.07i)12-s + (3.11 + 5.40i)13-s + (−0.412 − 0.714i)14-s + (2.44 − 4.23i)16-s − 6.10·17-s + (−3.51 − 3.41i)18-s − 5.57·19-s + ⋯
L(s)  = 1  + (−0.577 + 1.00i)2-s + (−0.614 − 0.788i)3-s + (−0.167 − 0.289i)4-s + (1.14 − 0.158i)6-s + (−0.0955 + 0.165i)7-s − 0.768·8-s + (−0.245 + 0.969i)9-s + (−0.467 + 0.809i)11-s + (−0.125 + 0.309i)12-s + (0.865 + 1.49i)13-s + (−0.110 − 0.191i)14-s + (0.611 − 1.05i)16-s − 1.47·17-s + (−0.828 − 0.805i)18-s − 1.27·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.912 - 0.409i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.912 - 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0992109 + 0.463132i\)
\(L(\frac12)\) \(\approx\) \(0.0992109 + 0.463132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.06 + 1.36i)T \)
5 \( 1 \)
good2 \( 1 + (0.816 - 1.41i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.252 - 0.437i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.55 - 2.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.11 - 5.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.10T + 17T^{2} \)
19 \( 1 + 5.57T + 19T^{2} \)
23 \( 1 + (-1.91 - 3.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.11 + 3.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.72T + 37T^{2} \)
41 \( 1 + (-2.72 - 4.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.663 - 1.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.85 + 3.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 + (1.44 + 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.42 + 2.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.20 + 2.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.54T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + (1.70 - 2.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.95 + 12.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + (5.53 - 9.59i)T + (-48.5 - 84.0i)T^{2} \)
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

−12.74855358999364205889152861418, −11.63720455942223543411001339725, −10.92118607409700936013742746416, −9.354278870507888920884495227458, −8.562204796245075773854548565554, −7.46559176117765110874632458722, −6.67505624575287669102583669779, −6.00975812739481164819786367814, −4.48735148278133742379249346408, −2.15472406361848420691502945178, 0.48602573921149477368503761848, 2.75575043710789330579402916837, 4.02026977336941297168911170536, 5.58315121206429082249634410935, 6.44235118202630253279761541326, 8.415882473383831573567071970873, 9.047608745313854660653200496001, 10.38814549975233398905095884525, 10.72242590566126662368118100365, 11.32607295930730487553846508279

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