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Properties

Label 2-15e2-45.23-c1-0-7
Degree $2$
Conductor $225$
Sign $0.484 + 0.874i$
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  + (0.430 − 1.60i)2-s + (−1.35 + 1.08i)3-s + (−0.661 − 0.382i)4-s + (1.15 + 2.63i)6-s + (1.73 + 0.465i)7-s + (1.45 − 1.45i)8-s + (0.661 − 2.92i)9-s + (3.12 − 1.80i)11-s + (1.30 − 0.198i)12-s + (1.27 − 0.342i)13-s + (1.49 − 2.59i)14-s + (−2.47 − 4.28i)16-s + (−0.277 − 0.277i)17-s + (−4.41 − 2.32i)18-s + 6.25i·19-s + ⋯
L(s)  = 1  + (0.304 − 1.13i)2-s + (−0.781 + 0.624i)3-s + (−0.330 − 0.191i)4-s + (0.471 + 1.07i)6-s + (0.656 + 0.175i)7-s + (0.513 − 0.513i)8-s + (0.220 − 0.975i)9-s + (0.942 − 0.544i)11-s + (0.377 − 0.0573i)12-s + (0.354 − 0.0950i)13-s + (0.399 − 0.692i)14-s + (−0.618 − 1.07i)16-s + (−0.0671 − 0.0671i)17-s + (−1.04 − 0.547i)18-s + 1.43i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14866 - 0.677021i\)
\(L(\frac12)\) \(\approx\) \(1.14866 - 0.677021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.35 - 1.08i)T \)
5 \( 1 \)
good2 \( 1 + (-0.430 + 1.60i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-1.73 - 0.465i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.27 + 0.342i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.277 + 0.277i)T + 17iT^{2} \)
19 \( 1 - 6.25iT - 19T^{2} \)
23 \( 1 + (0.579 + 2.16i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.56 + 2.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.42 - 4.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.55 - 5.55i)T - 37iT^{2} \)
41 \( 1 + (-1.29 - 0.744i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.10 + 4.10i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.02 + 3.82i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.48 - 7.48i)T - 53iT^{2} \)
59 \( 1 + (-0.279 + 0.483i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.90 - 10.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.01iT - 71T^{2} \)
73 \( 1 + (-1.29 - 1.29i)T + 73iT^{2} \)
79 \( 1 + (-6.96 + 4.02i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.560 - 0.150i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + (5.14 + 1.37i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.93241141009451060736681605210, −11.20155544246882622008451858405, −10.50439158043150642003764164403, −9.583721395123090415084349231528, −8.370207661895989345650639794896, −6.78168517371947728180189166795, −5.59945240510160999583513450912, −4.32257269703169355042133581743, −3.43540110291468377881803292703, −1.46803322556119830886126390119, 1.74383272779161177283296441205, 4.40638716128401856462819674694, 5.33462363762947447986614115786, 6.42359079603810495241911182587, 7.13751853537915012956413231277, 7.969339990791256169179567053072, 9.253628896875059394424749632883, 10.91603931469000032950664290422, 11.33483560638339040570898905246, 12.51539940678643916874293231548

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