Properties

Label 2-15e2-225.4-c3-0-8
Degree $2$
Conductor $225$
Sign $0.971 - 0.236i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.849 − 3.99i)2-s + (0.343 − 5.18i)3-s + (−7.92 + 3.52i)4-s + (5.82 + 9.54i)5-s + (−21.0 + 3.03i)6-s + (−17.1 + 9.91i)7-s + (1.62 + 2.23i)8-s + (−26.7 − 3.56i)9-s + (33.1 − 31.3i)10-s + (−0.600 + 0.127i)11-s + (15.5 + 42.3i)12-s + (−8.91 + 41.9i)13-s + (54.1 + 60.1i)14-s + (51.4 − 26.9i)15-s + (−38.8 + 43.1i)16-s + (47.0 + 64.6i)17-s + ⋯
L(s)  = 1  + (−0.300 − 1.41i)2-s + (0.0660 − 0.997i)3-s + (−0.990 + 0.441i)4-s + (0.521 + 0.853i)5-s + (−1.42 + 0.206i)6-s + (−0.926 + 0.535i)7-s + (0.0717 + 0.0987i)8-s + (−0.991 − 0.131i)9-s + (1.04 − 0.992i)10-s + (−0.0164 + 0.00350i)11-s + (0.374 + 1.01i)12-s + (−0.190 + 0.894i)13-s + (1.03 + 1.14i)14-s + (0.885 − 0.463i)15-s + (−0.607 + 0.674i)16-s + (0.670 + 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.971 - 0.236i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.579900 + 0.0695437i\)
\(L(\frac12)\) \(\approx\) \(0.579900 + 0.0695437i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.343 + 5.18i)T \)
5 \( 1 + (-5.82 - 9.54i)T \)
good2 \( 1 + (0.849 + 3.99i)T + (-7.30 + 3.25i)T^{2} \)
7 \( 1 + (17.1 - 9.91i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (0.600 - 0.127i)T + (1.21e3 - 541. i)T^{2} \)
13 \( 1 + (8.91 - 41.9i)T + (-2.00e3 - 893. i)T^{2} \)
17 \( 1 + (-47.0 - 64.6i)T + (-1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (-32.5 + 23.6i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (87.9 - 79.2i)T + (1.27e3 - 1.21e4i)T^{2} \)
29 \( 1 + (4.07 - 38.7i)T + (-2.38e4 - 5.07e3i)T^{2} \)
31 \( 1 + (33.0 + 314. i)T + (-2.91e4 + 6.19e3i)T^{2} \)
37 \( 1 + (-219. + 71.2i)T + (4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-64.7 - 13.7i)T + (6.29e4 + 2.80e4i)T^{2} \)
43 \( 1 + (358. - 207. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (541. + 56.9i)T + (1.01e5 + 2.15e4i)T^{2} \)
53 \( 1 + (206. - 284. i)T + (-4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (573. + 121. i)T + (1.87e5 + 8.35e4i)T^{2} \)
61 \( 1 + (-463. + 98.5i)T + (2.07e5 - 9.23e4i)T^{2} \)
67 \( 1 + (-760. + 79.9i)T + (2.94e5 - 6.25e4i)T^{2} \)
71 \( 1 + (23.7 + 17.2i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (467. + 152. i)T + (3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (34.2 - 326. i)T + (-4.82e5 - 1.02e5i)T^{2} \)
83 \( 1 + (555. - 1.24e3i)T + (-3.82e5 - 4.24e5i)T^{2} \)
89 \( 1 + (46.4 - 143. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (221. + 23.3i)T + (8.92e5 + 1.89e5i)T^{2} \)
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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

−11.70069686353491581030888557386, −11.14275425356051050573313385332, −9.772291701820729838847960952284, −9.443219721980688140420071895718, −7.968303662621186375893184539993, −6.63210466678420140349736840488, −5.93815475178987072612096458831, −3.57467337243545350191381832613, −2.59619633306404556873760318219, −1.63284372650998366633385865074, 0.25378121390407949498493053618, 3.15696310316317058562616819193, 4.78587055263147895034519275152, 5.55337334615023789882860155115, 6.55842683718532422460036534498, 7.88548323851341490470687138102, 8.708459444578090670009566388304, 9.756454483750443515540049068769, 10.13601697810603345445140534925, 11.78501629954773509575157709873

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