Properties

Label 2-15e2-225.196-c1-0-10
Degree $2$
Conductor $225$
Sign $0.940 - 0.338i$
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  + (0.797 + 0.885i)2-s + (−1.56 − 0.733i)3-s + (0.0604 − 0.575i)4-s + (1.30 + 1.81i)5-s + (−0.602 − 1.97i)6-s + (0.157 − 0.272i)7-s + (2.48 − 1.80i)8-s + (1.92 + 2.30i)9-s + (−0.566 + 2.60i)10-s + (1.93 + 2.14i)11-s + (−0.516 + 0.858i)12-s + (4.36 − 4.85i)13-s + (0.367 − 0.0781i)14-s + (−0.719 − 3.80i)15-s + (2.45 + 0.521i)16-s + (0.794 − 0.577i)17-s + ⋯
L(s)  = 1  + (0.564 + 0.626i)2-s + (−0.906 − 0.423i)3-s + (0.0302 − 0.287i)4-s + (0.584 + 0.811i)5-s + (−0.245 − 0.806i)6-s + (0.0595 − 0.103i)7-s + (0.879 − 0.638i)8-s + (0.641 + 0.766i)9-s + (−0.179 + 0.823i)10-s + (0.583 + 0.647i)11-s + (−0.149 + 0.247i)12-s + (1.21 − 1.34i)13-s + (0.0982 − 0.0208i)14-s + (−0.185 − 0.982i)15-s + (0.613 + 0.130i)16-s + (0.192 − 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.940 - 0.338i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.940 - 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45054 + 0.253220i\)
\(L(\frac12)\) \(\approx\) \(1.45054 + 0.253220i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 + 0.733i)T \)
5 \( 1 + (-1.30 - 1.81i)T \)
good2 \( 1 + (-0.797 - 0.885i)T + (-0.209 + 1.98i)T^{2} \)
7 \( 1 + (-0.157 + 0.272i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.93 - 2.14i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-4.36 + 4.85i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (-0.794 + 0.577i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.88 - 4.27i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.28 + 0.272i)T + (21.0 - 9.35i)T^{2} \)
29 \( 1 + (3.94 - 1.75i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (7.91 + 3.52i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-0.00738 - 0.0227i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.06 + 3.40i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (1.34 - 2.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.76 - 2.12i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (3.44 + 2.50i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.46 + 3.84i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (4.95 + 5.50i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (2.27 + 1.01i)T + (44.8 + 49.7i)T^{2} \)
71 \( 1 + (-9.63 - 7.00i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.283 - 0.872i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.3 - 4.62i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-1.03 - 9.88i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 + (-3.78 + 11.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.75 - 3.00i)T + (64.9 - 72.0i)T^{2} \)
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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

−12.68805904856569402679844836819, −11.04747751980182670943521411836, −10.66475165602129073135317847377, −9.679735711954971087205908912211, −7.85462267391715482288484715704, −6.86075133844797830731214218668, −6.07046247639122664383866718575, −5.44630874779649818838713589039, −3.93607526901711181379964232576, −1.65064791605194197304253738365, 1.67777899600964975604989546505, 3.77790065659370990435504570904, 4.58484336433911971142553325720, 5.76640892117427536103499700401, 6.78500717750421154326544100337, 8.616595112729179873715562725994, 9.239921532746259974039793312548, 10.73669845571165229512577886658, 11.30746778569059809339962997235, 12.10662568912641666239290045990

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