Properties

Label 2-15e2-25.6-c1-0-2
Degree $2$
Conductor $225$
Sign $-0.977 - 0.211i$
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  + (0.830 + 2.55i)2-s + (−4.22 + 3.06i)4-s + (1.34 + 1.78i)5-s + 1.68·7-s + (−7.00 − 5.08i)8-s + (−3.43 + 4.92i)10-s + (−0.333 − 1.02i)11-s + (0.827 − 2.54i)13-s + (1.40 + 4.31i)14-s + (3.95 − 12.1i)16-s + (−3.18 − 2.31i)17-s + (0.952 + 0.692i)19-s + (−11.1 − 3.39i)20-s + (2.34 − 1.70i)22-s + (1.25 + 3.86i)23-s + ⋯
L(s)  = 1  + (0.587 + 1.80i)2-s + (−2.11 + 1.53i)4-s + (0.603 + 0.797i)5-s + 0.637·7-s + (−2.47 − 1.79i)8-s + (−1.08 + 1.55i)10-s + (−0.100 − 0.309i)11-s + (0.229 − 0.706i)13-s + (0.374 + 1.15i)14-s + (0.989 − 3.04i)16-s + (−0.771 − 0.560i)17-s + (0.218 + 0.158i)19-s + (−2.49 − 0.759i)20-s + (0.499 − 0.363i)22-s + (0.261 + 0.805i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167973 + 1.56853i\)
\(L(\frac12)\) \(\approx\) \(0.167973 + 1.56853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.34 - 1.78i)T \)
good2 \( 1 + (-0.830 - 2.55i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 - 1.68T + 7T^{2} \)
11 \( 1 + (0.333 + 1.02i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.827 + 2.54i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.18 + 2.31i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.952 - 0.692i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.25 - 3.86i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.81 + 3.50i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.74 - 4.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.41 - 4.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.47 + 10.7i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.58T + 43T^{2} \)
47 \( 1 + (1.55 - 1.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.06 - 1.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.412 - 1.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.24 + 6.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (10.0 + 7.31i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.84 - 3.51i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.01 + 3.13i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.23 + 2.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.17 - 5.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.77 + 14.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.71 - 6.32i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.31465052551192553586618359313, −12.00546192700923978676300140110, −10.69641063755187668286329349151, −9.420973923587110601307010678002, −8.364331965192888424246128422774, −7.48501263148728286206062490058, −6.54117889508960053138283785296, −5.66437697876482485839554649080, −4.68769327008295597794951442463, −3.15668192412204880425547358634, 1.35034081090724117134563503197, 2.51829339386102864612133100094, 4.30705291871504541749918555371, 4.86124152941960261876299469725, 6.19097050102803087744716041313, 8.404500561959597408135996437398, 9.185197993114260891163760599648, 10.10276200968876860987680649800, 11.00072523123994899910912938685, 11.83182050887870547202363707688

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