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Properties

Label 2-15e2-45.34-c1-0-12
Degree $2$
Conductor $225$
Sign $0.00188 + 0.999i$
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  + (−0.409 − 0.236i)2-s + (1.64 − 0.544i)3-s + (−0.888 − 1.53i)4-s + (−0.802 − 0.165i)6-s + (−2.21 − 1.28i)7-s + 1.78i·8-s + (2.40 − 1.79i)9-s + (3.08 − 5.34i)11-s + (−2.29 − 2.04i)12-s + (−1.84 + 1.06i)13-s + (0.606 + 1.05i)14-s + (−1.35 + 2.34i)16-s + 3.16i·17-s + (−1.41 + 0.164i)18-s − 0.356·19-s + ⋯
L(s)  = 1  + (−0.289 − 0.167i)2-s + (0.949 − 0.314i)3-s + (−0.444 − 0.769i)4-s + (−0.327 − 0.0676i)6-s + (−0.838 − 0.484i)7-s + 0.631i·8-s + (0.802 − 0.597i)9-s + (0.929 − 1.61i)11-s + (−0.663 − 0.590i)12-s + (−0.512 + 0.295i)13-s + (0.162 + 0.280i)14-s + (−0.338 + 0.585i)16-s + 0.768i·17-s + (−0.332 + 0.0388i)18-s − 0.0817·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00188 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00188 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.00188 + 0.999i$
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: \((2,\ 225,\ (\ :1/2),\ 0.00188 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.852768 - 0.851162i\)
\(L(\frac12)\) \(\approx\) \(0.852768 - 0.851162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 + 0.544i)T \)
5 \( 1 \)
good2 \( 1 + (0.409 + 0.236i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (2.21 + 1.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.08 + 5.34i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.84 - 1.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.16iT - 17T^{2} \)
19 \( 1 + 0.356T + 19T^{2} \)
23 \( 1 + (-3.64 + 2.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.843 + 1.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.12 - 7.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.63iT - 37T^{2} \)
41 \( 1 + (-1.36 - 2.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.64 + 3.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.89 - 5.71i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.43iT - 53T^{2} \)
59 \( 1 + (-5.10 - 8.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.00549 + 0.00952i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.851 + 0.491i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.43T + 71T^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 + (4.73 - 8.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.02 + 5.20i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.26T + 89T^{2} \)
97 \( 1 + (6.24 + 3.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.06623355648234813475733811674, −10.76552487917367694209300787298, −9.946374832763376910788834035025, −8.971840044944014036228526265934, −8.449725099539450824152222967250, −6.89646447876743806484905261260, −6.03831641392424735186967500558, −4.27480773071864363393388034694, −3.04012209088300663025575580994, −1.12791832287013493381768540324, 2.52948300933824166509496340635, 3.76123392970715052758414092128, 4.83702100835251148014575061224, 6.85810321418898862724445065689, 7.52891879155706592888803758414, 8.739604046269946432075563160387, 9.546194585702069902733297526191, 9.900010073674263831451961481085, 11.78540432346378101734101540032, 12.65528697431924105514581113331

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