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Properties

Label 2-15e2-225.79-c1-0-14
Degree $2$
Conductor $225$
Sign $0.893 - 0.449i$
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  + (0.894 − 0.805i)2-s + (0.487 + 1.66i)3-s + (−0.0574 + 0.546i)4-s + (2.23 + 0.0854i)5-s + (1.77 + 1.09i)6-s + (−2.32 − 1.33i)7-s + (1.80 + 2.48i)8-s + (−2.52 + 1.61i)9-s + (2.06 − 1.72i)10-s + (−0.941 − 1.04i)11-s + (−0.936 + 0.170i)12-s + (−0.120 − 0.108i)13-s + (−3.15 + 0.670i)14-s + (0.946 + 3.75i)15-s + (2.54 + 0.540i)16-s + (0.113 + 0.156i)17-s + ⋯
L(s)  = 1  + (0.632 − 0.569i)2-s + (0.281 + 0.959i)3-s + (−0.0287 + 0.273i)4-s + (0.999 + 0.0381i)5-s + (0.724 + 0.447i)6-s + (−0.877 − 0.506i)7-s + (0.638 + 0.878i)8-s + (−0.841 + 0.539i)9-s + (0.654 − 0.545i)10-s + (−0.283 − 0.315i)11-s + (−0.270 + 0.0493i)12-s + (−0.0335 − 0.0301i)13-s + (−0.843 + 0.179i)14-s + (0.244 + 0.969i)15-s + (0.635 + 0.135i)16-s + (0.0275 + 0.0378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82863 + 0.433761i\)
\(L(\frac12)\) \(\approx\) \(1.82863 + 0.433761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.487 - 1.66i)T \)
5 \( 1 + (-2.23 - 0.0854i)T \)
good2 \( 1 + (-0.894 + 0.805i)T + (0.209 - 1.98i)T^{2} \)
7 \( 1 + (2.32 + 1.33i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.941 + 1.04i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (0.120 + 0.108i)T + (1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.113 - 0.156i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.07 + 4.41i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.22 + 5.77i)T + (-21.0 + 9.35i)T^{2} \)
29 \( 1 + (5.56 - 2.47i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (0.549 + 0.244i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-0.888 + 0.288i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.78 + 4.20i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (5.91 + 3.41i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0890 - 0.199i)T + (-31.4 + 34.9i)T^{2} \)
53 \( 1 + (7.64 - 10.5i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.97 - 4.41i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-9.43 - 10.4i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (-2.62 + 5.90i)T + (-44.8 - 49.7i)T^{2} \)
71 \( 1 + (-2.90 - 2.11i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (11.0 + 3.59i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-12.4 + 5.53i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (14.5 - 1.53i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + (3.38 - 10.4i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.26 + 2.83i)T + (-64.9 + 72.0i)T^{2} \)
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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

−12.47281654386360391044260961952, −11.20204046029837514008649048567, −10.46831064834342038538752127429, −9.583808038815354699920863322821, −8.699202531695210927947275752277, −7.26999500576622121342321194892, −5.75150361321226197650328309302, −4.73996676725739380209343818171, −3.47152114306971204683205283566, −2.59874057061387718334390554107, 1.68495837120750904585382030389, 3.30473000318621512334834219164, 5.35296032844575208321466848783, 5.95716348688421495948489691697, 6.84189141905570871772006212858, 7.88774835637373609621854136353, 9.550751754057219875437868470908, 9.763553848965553864610740647993, 11.47121963612913977847632036060, 12.73672471379779131149531527870

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