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Properties

Label 2-15e2-225.2-c1-0-1
Degree $2$
Conductor $225$
Sign $0.831 - 0.556i$
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  + (−1.88 − 1.52i)2-s + (−1.53 − 0.804i)3-s + (0.811 + 3.81i)4-s + (−0.154 − 2.23i)5-s + (1.66 + 3.86i)6-s + (−3.12 + 0.838i)7-s + (2.09 − 4.11i)8-s + (1.70 + 2.46i)9-s + (−3.11 + 4.44i)10-s + (−3.96 + 0.416i)11-s + (1.82 − 6.50i)12-s + (0.956 + 1.18i)13-s + (7.19 + 3.20i)14-s + (−1.55 + 3.54i)15-s + (−3.12 + 1.39i)16-s + (4.14 + 2.11i)17-s + ⋯
L(s)  = 1  + (−1.33 − 1.08i)2-s + (−0.885 − 0.464i)3-s + (0.405 + 1.90i)4-s + (−0.0691 − 0.997i)5-s + (0.680 + 1.57i)6-s + (−1.18 + 0.316i)7-s + (0.742 − 1.45i)8-s + (0.568 + 0.822i)9-s + (−0.986 + 1.40i)10-s + (−1.19 + 0.125i)11-s + (0.526 − 1.87i)12-s + (0.265 + 0.327i)13-s + (1.92 + 0.855i)14-s + (−0.401 + 0.915i)15-s + (−0.782 + 0.348i)16-s + (1.00 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.831 - 0.556i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.831 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136006 + 0.0413174i\)
\(L(\frac12)\) \(\approx\) \(0.136006 + 0.0413174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 + 0.804i)T \)
5 \( 1 + (0.154 + 2.23i)T \)
good2 \( 1 + (1.88 + 1.52i)T + (0.415 + 1.95i)T^{2} \)
7 \( 1 + (3.12 - 0.838i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.96 - 0.416i)T + (10.7 - 2.28i)T^{2} \)
13 \( 1 + (-0.956 - 1.18i)T + (-2.70 + 12.7i)T^{2} \)
17 \( 1 + (-4.14 - 2.11i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.34 + 0.436i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.610 + 1.59i)T + (-17.0 + 15.3i)T^{2} \)
29 \( 1 + (-5.15 - 5.72i)T + (-3.03 + 28.8i)T^{2} \)
31 \( 1 + (-0.568 + 0.631i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-1.50 - 9.48i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (11.0 + 1.16i)T + (40.1 + 8.52i)T^{2} \)
43 \( 1 + (-2.09 - 7.82i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.56 + 0.291i)T + (46.7 + 4.91i)T^{2} \)
53 \( 1 + (9.89 - 5.04i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.511 - 4.86i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (0.958 + 9.11i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-0.0988 + 0.00518i)T + (66.6 - 7.00i)T^{2} \)
71 \( 1 + (6.77 + 2.20i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.165 + 1.04i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.02 - 1.82i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (-7.09 - 10.9i)T + (-33.7 + 75.8i)T^{2} \)
89 \( 1 + (4.64 - 3.37i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.736 + 14.0i)T + (-96.4 - 10.1i)T^{2} \)
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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

−12.34487865183725778576497327131, −11.33554847234118204982743111254, −10.23363218931199214240114742823, −9.746747582417436729720008836867, −8.516690934631855084602405231762, −7.75470738345689677439964186078, −6.34548169078203973252101138309, −5.00537132893284508154885155888, −3.06451570417243867972609517702, −1.37476794510245418185732140194, 0.21350158727012729868452823323, 3.33705624823361449054179995517, 5.44785935698651748778657630475, 6.26048484922400017880605129789, 7.10656437697035711879843520842, 7.950525602804128969242189576115, 9.480984666496535250684333144312, 10.18528726584473625221987499393, 10.53987839102647228999206028090, 11.80294424180567591633364239377

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