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Properties

Label 2-15e2-45.34-c1-0-11
Degree $2$
Conductor $225$
Sign $0.792 - 0.610i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.28 + 1.31i)2-s + (0.238 − 1.71i)3-s + (2.46 + 4.27i)4-s + (2.80 − 3.59i)6-s + (−1.55 − 0.898i)7-s + 7.73i·8-s + (−2.88 − 0.817i)9-s + (−0.904 + 1.56i)11-s + (7.92 − 3.21i)12-s + (1.70 − 0.985i)13-s + (−2.36 − 4.09i)14-s + (−5.24 + 9.08i)16-s − 4.80i·17-s + (−5.50 − 5.66i)18-s − 2.96·19-s + ⋯
L(s)  = 1  + (1.61 + 0.931i)2-s + (0.137 − 0.990i)3-s + (1.23 + 2.13i)4-s + (1.14 − 1.46i)6-s + (−0.588 − 0.339i)7-s + 2.73i·8-s + (−0.962 − 0.272i)9-s + (−0.272 + 0.472i)11-s + (2.28 − 0.928i)12-s + (0.473 − 0.273i)13-s + (−0.632 − 1.09i)14-s + (−1.31 + 2.27i)16-s − 1.16i·17-s + (−1.29 − 1.33i)18-s − 0.680·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.792 - 0.610i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.792 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53085 + 0.861991i\)
\(L(\frac12)\) \(\approx\) \(2.53085 + 0.861991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.238 + 1.71i)T \)
5 \( 1 \)
good2 \( 1 + (-2.28 - 1.31i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.55 + 0.898i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.904 - 1.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.70 + 0.985i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.80iT - 17T^{2} \)
19 \( 1 + 2.96T + 19T^{2} \)
23 \( 1 + (1.50 - 0.866i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.68 - 6.38i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.31 - 2.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.6iT - 37T^{2} \)
41 \( 1 + (-1.23 - 2.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.30 - 3.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.44 - 3.14i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.72iT - 53T^{2} \)
59 \( 1 + (-5.51 - 9.54i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.33 + 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.88 - 4.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + 3.58iT - 73T^{2} \)
79 \( 1 + (-1.05 + 1.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.951 + 0.549i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (3.31 + 1.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.78515609351411690062491185679, −11.97269741236369948599443939201, −10.88350995811472105189962659718, −9.026531209513324832253450598925, −7.70996072009093020450858093542, −7.11153161677681343483277662200, −6.21101019512445987250701341432, −5.24100276876766209872398357931, −3.79923968309837460661318450403, −2.59648827425710391273386170049, 2.39325361301744328260467100711, 3.59089617788130853068209135670, 4.35349495779173586102113547120, 5.67681973233967586888968085615, 6.28633364335107827717451763386, 8.441186834727304058259355583872, 9.726963002330724381157688760200, 10.52199796656251235192487841609, 11.26479918996626588455673958944, 12.15926130406674185632830789241

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