Properties

Label 2-15e2-45.32-c1-0-14
Degree $2$
Conductor $225$
Sign $0.984 + 0.177i$
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  + (2.24 + 0.601i)2-s + (−0.173 − 1.72i)3-s + (2.93 + 1.69i)4-s + (0.647 − 3.97i)6-s + (0.201 − 0.751i)7-s + (2.29 + 2.29i)8-s + (−2.93 + 0.597i)9-s + (−0.220 + 0.127i)11-s + (2.41 − 5.36i)12-s + (0.992 + 3.70i)13-s + (0.903 − 1.56i)14-s + (0.367 + 0.636i)16-s + (−3.93 + 3.93i)17-s + (−6.95 − 0.427i)18-s + 0.440i·19-s + ⋯
L(s)  = 1  + (1.58 + 0.425i)2-s + (−0.100 − 0.994i)3-s + (1.46 + 0.848i)4-s + (0.264 − 1.62i)6-s + (0.0761 − 0.284i)7-s + (0.809 + 0.809i)8-s + (−0.979 + 0.199i)9-s + (−0.0663 + 0.0383i)11-s + (0.697 − 1.54i)12-s + (0.275 + 1.02i)13-s + (0.241 − 0.418i)14-s + (0.0918 + 0.159i)16-s + (−0.953 + 0.953i)17-s + (−1.63 − 0.100i)18-s + 0.101i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53698 - 0.227210i\)
\(L(\frac12)\) \(\approx\) \(2.53698 - 0.227210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.173 + 1.72i)T \)
5 \( 1 \)
good2 \( 1 + (-2.24 - 0.601i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-0.201 + 0.751i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.220 - 0.127i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.992 - 3.70i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.93 - 3.93i)T - 17iT^{2} \)
19 \( 1 - 0.440iT - 19T^{2} \)
23 \( 1 + (-3.42 + 0.917i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.76 + 4.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0971 - 0.168i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.123 - 0.123i)T + 37iT^{2} \)
41 \( 1 + (3.88 + 2.24i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.33 + 0.357i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.17 + 1.11i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.938 + 0.938i)T + 53iT^{2} \)
59 \( 1 + (4.02 - 6.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.44 + 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.9 + 3.47i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.15iT - 71T^{2} \)
73 \( 1 + (-9.18 + 9.18i)T - 73iT^{2} \)
79 \( 1 + (-11.9 + 6.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.39 + 5.20i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 0.285T + 89T^{2} \)
97 \( 1 + (-2.34 + 8.73i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.50369119048513052794665187952, −11.66701740791377203081157634270, −10.86176691015259898725400457281, −9.018725320015374766187883920744, −7.77318618744645989959144349369, −6.73632325418328369412375932111, −6.17008646090655856619332606485, −4.89955566838860141138499768875, −3.71891586839561167832284671242, −2.13387744146470272935746528066, 2.68455126968871629347101327339, 3.66187296808383412624866064450, 4.91186929975716370112056249518, 5.47435673431666687160071722471, 6.72654755633397998272410626495, 8.474613704917700239084984779660, 9.607969259126201067422355791739, 10.86996534071823485425051000403, 11.27221405188945832041693867710, 12.34172991354759908267307930874

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