Properties

Label 2-15e2-225.184-c1-0-23
Degree $2$
Conductor $225$
Sign $-0.775 + 0.630i$
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  + (0.649 + 0.0682i)2-s + (−1.09 + 1.34i)3-s + (−1.53 − 0.327i)4-s + (−1.81 − 1.30i)5-s + (−0.803 + 0.795i)6-s + (0.401 − 0.231i)7-s + (−2.21 − 0.721i)8-s + (−0.594 − 2.94i)9-s + (−1.09 − 0.968i)10-s + (0.190 − 1.81i)11-s + (2.12 − 1.70i)12-s + (−5.26 + 0.553i)13-s + (0.276 − 0.123i)14-s + (3.73 − 1.01i)15-s + (1.48 + 0.659i)16-s + (−3.97 − 1.29i)17-s + ⋯
L(s)  = 1  + (0.459 + 0.0482i)2-s + (−0.633 + 0.773i)3-s + (−0.769 − 0.163i)4-s + (−0.813 − 0.581i)5-s + (−0.328 + 0.324i)6-s + (0.151 − 0.0876i)7-s + (−0.784 − 0.254i)8-s + (−0.198 − 0.980i)9-s + (−0.345 − 0.306i)10-s + (0.0575 − 0.547i)11-s + (0.613 − 0.492i)12-s + (−1.45 + 0.153i)13-s + (0.0739 − 0.0329i)14-s + (0.965 − 0.261i)15-s + (0.370 + 0.164i)16-s + (−0.963 − 0.312i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.775 + 0.630i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.775 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0686984 - 0.193398i\)
\(L(\frac12)\) \(\approx\) \(0.0686984 - 0.193398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.09 - 1.34i)T \)
5 \( 1 + (1.81 + 1.30i)T \)
good2 \( 1 + (-0.649 - 0.0682i)T + (1.95 + 0.415i)T^{2} \)
7 \( 1 + (-0.401 + 0.231i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.190 + 1.81i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (5.26 - 0.553i)T + (12.7 - 2.70i)T^{2} \)
17 \( 1 + (3.97 + 1.29i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.455 - 1.40i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.266 - 0.598i)T + (-15.3 + 17.0i)T^{2} \)
29 \( 1 + (-0.256 + 0.284i)T + (-3.03 - 28.8i)T^{2} \)
31 \( 1 + (2.55 + 2.83i)T + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (-5.88 - 8.09i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.842 + 8.01i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (6.11 - 3.52i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.90 + 6.21i)T + (4.91 + 46.7i)T^{2} \)
53 \( 1 + (0.545 - 0.177i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.18 + 11.2i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + (0.0903 - 0.859i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (-5.70 + 5.13i)T + (7.00 - 66.6i)T^{2} \)
71 \( 1 + (4.37 + 13.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.25 + 7.23i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-6.20 + 6.89i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-2.80 - 13.2i)T + (-75.8 + 33.7i)T^{2} \)
89 \( 1 + (-4.64 - 3.37i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (10.3 + 9.28i)T + (10.1 + 96.4i)T^{2} \)
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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

−11.92034729372842845039681677508, −11.09230462473793181605732972810, −9.811188666075827977390397428297, −9.116785277874368161190626798296, −8.027412111368403066807426974615, −6.48651410623530475805066739063, −5.10616782949266992480871500755, −4.62118617263698705072056757175, −3.50743150768134827645874982626, −0.15592466992740019900939798084, 2.60040179575250193055915232699, 4.30103556922755963420372752764, 5.17520708266874946623392393578, 6.60927260914540391620231591915, 7.50052780272952421554237229466, 8.453734645741558739476666282500, 9.816369979932918668217318281248, 11.02907701785870432734504929854, 11.87490428257244695836888503212, 12.59829355281859023217806464692

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