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Properties

Label 2-15e2-5.3-c4-0-19
Degree $2$
Conductor $225$
Sign $0.525 + 0.850i$
Analytic cond. $23.2582$
Root an. cond. $4.82267$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.30 − 3.30i)2-s − 5.84i·4-s + (33.1 − 33.1i)7-s + (33.5 + 33.5i)8-s − 55.3·11-s + (161. + 161. i)13-s − 219. i·14-s + 315.·16-s + (278. − 278. i)17-s + 179. i·19-s + (−182. + 182. i)22-s + (−398. − 398. i)23-s + 1.07e3·26-s + (−193. − 193. i)28-s − 547. i·29-s + ⋯
L(s)  = 1  + (0.826 − 0.826i)2-s − 0.365i·4-s + (0.676 − 0.676i)7-s + (0.524 + 0.524i)8-s − 0.457·11-s + (0.958 + 0.958i)13-s − 1.11i·14-s + 1.23·16-s + (0.965 − 0.965i)17-s + 0.498i·19-s + (−0.377 + 0.377i)22-s + (−0.752 − 0.752i)23-s + 1.58·26-s + (−0.247 − 0.247i)28-s − 0.650i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(23.2582\)
Root analytic conductor: \(4.82267\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :2),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.509344957\)
\(L(\frac12)\) \(\approx\) \(3.509344957\)
\(L(3)\) 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 \( 1 \)
good2 \( 1 + (-3.30 + 3.30i)T - 16iT^{2} \)
7 \( 1 + (-33.1 + 33.1i)T - 2.40e3iT^{2} \)
11 \( 1 + 55.3T + 1.46e4T^{2} \)
13 \( 1 + (-161. - 161. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-278. + 278. i)T - 8.35e4iT^{2} \)
19 \( 1 - 179. iT - 1.30e5T^{2} \)
23 \( 1 + (398. + 398. i)T + 2.79e5iT^{2} \)
29 \( 1 + 547. iT - 7.07e5T^{2} \)
31 \( 1 - 1.53e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.66e3 + 1.66e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 307.T + 2.82e6T^{2} \)
43 \( 1 + (104. + 104. i)T + 3.41e6iT^{2} \)
47 \( 1 + (346. - 346. i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.02e3 + 2.02e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 2.85e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.05e3T + 1.38e7T^{2} \)
67 \( 1 + (3.75e3 - 3.75e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.42e3T + 2.54e7T^{2} \)
73 \( 1 + (813. + 813. i)T + 2.83e7iT^{2} \)
79 \( 1 - 4.85e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.31e3 + 1.31e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 5.18e3iT - 6.27e7T^{2} \)
97 \( 1 + (3.49e3 - 3.49e3i)T - 8.85e7iT^{2} \)
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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

−11.54732851941957759558373224133, −10.78161526033274485183844726136, −9.828357063166986677878408926373, −8.310824417953207435416675563812, −7.53349567675609078341672617129, −6.00321153879035799589550551748, −4.67938008087282857405080999282, −3.93239174201237433102766955406, −2.55883133983498877921029442982, −1.17598293090456730018618480934, 1.31507940831095281010337193121, 3.22662499145853306147369668345, 4.62428100242563282904142318084, 5.61461119493455474941683672478, 6.26149694854711837224983402248, 7.76991249108865868247502742261, 8.335194081494378176646535214084, 9.894354372083985546191315586636, 10.79054868696523020745866168907, 11.95545548300864286612832235071

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