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Properties

Label 2-15e2-25.16-c1-0-10
Degree $2$
Conductor $225$
Sign $-0.841 + 0.540i$
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  + (−0.114 − 0.0832i)2-s + (−0.611 − 1.88i)4-s + (−2.14 − 0.617i)5-s − 0.858·7-s + (−0.174 + 0.536i)8-s + (0.194 + 0.249i)10-s + (−2.97 − 2.16i)11-s + (−3.70 + 2.69i)13-s + (0.0983 + 0.0714i)14-s + (−3.13 + 2.28i)16-s + (1.63 − 5.04i)17-s + (1.96 − 6.05i)19-s + (0.151 + 4.42i)20-s + (0.161 + 0.495i)22-s + (2.76 + 2.01i)23-s + ⋯
L(s)  = 1  + (−0.0810 − 0.0588i)2-s + (−0.305 − 0.941i)4-s + (−0.961 − 0.276i)5-s − 0.324·7-s + (−0.0616 + 0.189i)8-s + (0.0616 + 0.0789i)10-s + (−0.897 − 0.652i)11-s + (−1.02 + 0.746i)13-s + (0.0262 + 0.0191i)14-s + (−0.784 + 0.570i)16-s + (0.397 − 1.22i)17-s + (0.451 − 1.38i)19-s + (0.0338 + 0.989i)20-s + (0.0343 + 0.105i)22-s + (0.577 + 0.419i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150465 - 0.512664i\)
\(L(\frac12)\) \(\approx\) \(0.150465 - 0.512664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.14 + 0.617i)T \)
good2 \( 1 + (0.114 + 0.0832i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + 0.858T + 7T^{2} \)
11 \( 1 + (2.97 + 2.16i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.70 - 2.69i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.63 + 5.04i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.96 + 6.05i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.76 - 2.01i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.15 + 3.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.387 + 1.19i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.02 + 4.37i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.04 - 1.48i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.37T + 43T^{2} \)
47 \( 1 + (-2.62 - 8.08i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.725 - 2.23i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.6 + 7.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.37 + 6.08i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.03 + 3.18i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.33 - 4.11i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.34 - 5.33i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.00 + 3.08i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.28 + 7.03i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (12.5 + 9.10i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.10 + 9.54i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.56825206698401694438187043268, −11.10057614975429717308584452410, −9.718580419524756919821102141772, −9.156870282993344551791299266903, −7.83113035344289410379999174131, −6.86068989732090563412060094700, −5.34285538646958092336845905193, −4.58438819540406549188008404805, −2.82687652019421388834920781073, −0.44626389904788896235768033506, 2.88190064410123069906962777442, 3.91610362688519829335797983764, 5.19476684055802258218887744843, 6.92132700942258169735154405023, 7.81592812814785960801080229309, 8.331450083190104302225048720733, 9.835306479228075657834616183377, 10.62533421498576872615202609214, 12.04318109946238430270125248040, 12.46879736014730590987615672446

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