Properties

Label 2-15e2-225.16-c1-0-10
Degree $2$
Conductor $225$
Sign $0.854 - 0.520i$
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  + (−2.53 + 1.13i)2-s + (1.12 − 1.31i)3-s + (3.82 − 4.25i)4-s + (1.17 + 1.90i)5-s + (−1.36 + 4.61i)6-s + (−0.665 + 1.15i)7-s + (−3.19 + 9.83i)8-s + (−0.478 − 2.96i)9-s + (−5.13 − 3.50i)10-s + (1.76 − 0.787i)11-s + (−1.30 − 9.82i)12-s + (3.99 + 1.78i)13-s + (0.386 − 3.68i)14-s + (3.82 + 0.587i)15-s + (−1.80 − 17.2i)16-s + (0.413 − 1.27i)17-s + ⋯
L(s)  = 1  + (−1.79 + 0.799i)2-s + (0.648 − 0.761i)3-s + (1.91 − 2.12i)4-s + (0.525 + 0.850i)5-s + (−0.555 + 1.88i)6-s + (−0.251 + 0.435i)7-s + (−1.13 + 3.47i)8-s + (−0.159 − 0.987i)9-s + (−1.62 − 1.10i)10-s + (0.533 − 0.237i)11-s + (−0.377 − 2.83i)12-s + (1.10 + 0.493i)13-s + (0.103 − 0.983i)14-s + (0.988 + 0.151i)15-s + (−0.452 − 4.30i)16-s + (0.100 − 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.854 - 0.520i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.854 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735934 + 0.206416i\)
\(L(\frac12)\) \(\approx\) \(0.735934 + 0.206416i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.12 + 1.31i)T \)
5 \( 1 + (-1.17 - 1.90i)T \)
good2 \( 1 + (2.53 - 1.13i)T + (1.33 - 1.48i)T^{2} \)
7 \( 1 + (0.665 - 1.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.76 + 0.787i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-3.99 - 1.78i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (-0.413 + 1.27i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.593 - 1.82i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.0712 + 0.677i)T + (-22.4 - 4.78i)T^{2} \)
29 \( 1 + (-3.36 - 0.715i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (-8.56 + 1.81i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-1.12 + 0.820i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.65 - 0.738i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (-0.906 + 1.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.43 + 0.942i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (1.29 + 3.98i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.4 + 5.09i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (7.41 - 3.30i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (5.18 - 1.10i)T + (61.2 - 27.2i)T^{2} \)
71 \( 1 + (0.998 + 3.07i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.916 - 0.665i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (10.3 + 2.19i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (6.90 + 7.66i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (-13.1 - 9.53i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (17.6 + 3.74i)T + (88.6 + 39.4i)T^{2} \)
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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

−11.95707497656106748790072456903, −11.06308232464926707527218963937, −9.981782136757247895411209523797, −9.167418749255765283079354188721, −8.448020438374675577918218883070, −7.45728597666464319724029309110, −6.36920565623371573900301666935, −6.14199054806136809254733713735, −2.81879607841731583225215884160, −1.48159401213829295520469426509, 1.33458726003592504527207411197, 2.91904348759050181387872014537, 4.20456469735159727726686831788, 6.36030500380415898724664796072, 7.84390886810607512166567209551, 8.603610566633317540315051782670, 9.257083284784851110470124756785, 10.09223803047761516110589926313, 10.69434165245849459267388064438, 11.80913533023356676058231751141

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