Properties

Label 2-15e2-45.4-c3-0-20
Degree $2$
Conductor $225$
Sign $0.998 - 0.0524i$
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  + (2.92 − 1.68i)2-s + (−4.97 − 1.5i)3-s + (1.68 − 2.92i)4-s + (−17.0 + 4.00i)6-s + (−1.40 + 0.813i)7-s + 15.6i·8-s + (22.5 + 14.9i)9-s + (16.4 + 28.4i)11-s + (−12.7 + 12i)12-s + (28.5 + 16.5i)13-s + (−2.74 + 4.75i)14-s + (39.8 + 68.9i)16-s − 110. i·17-s + (90.8 + 5.64i)18-s + 54.3·19-s + ⋯
L(s)  = 1  + (1.03 − 0.596i)2-s + (−0.957 − 0.288i)3-s + (0.210 − 0.365i)4-s + (−1.16 + 0.272i)6-s + (−0.0761 + 0.0439i)7-s + 0.689i·8-s + (0.833 + 0.552i)9-s + (0.450 + 0.780i)11-s + (−0.307 + 0.288i)12-s + (0.610 + 0.352i)13-s + (−0.0523 + 0.0907i)14-s + (0.621 + 1.07i)16-s − 1.57i·17-s + (1.18 + 0.0739i)18-s + 0.655·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.998 - 0.0524i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.998 - 0.0524i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.20317 + 0.0578411i\)
\(L(\frac12)\) \(\approx\) \(2.20317 + 0.0578411i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.97 + 1.5i)T \)
5 \( 1 \)
good2 \( 1 + (-2.92 + 1.68i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (1.40 - 0.813i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-16.4 - 28.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-28.5 - 16.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 110. iT - 4.91e3T^{2} \)
19 \( 1 - 54.3T + 6.85e3T^{2} \)
23 \( 1 + (-58.4 - 33.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-137. - 237. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 347. iT - 5.06e4T^{2} \)
41 \( 1 + (145. - 252. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-174. + 100. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-417. + 241. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 175. iT - 1.48e5T^{2} \)
59 \( 1 + (91.6 - 158. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (218. + 378. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (720. + 415. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 118.T + 3.57e5T^{2} \)
73 \( 1 - 183. iT - 3.89e5T^{2} \)
79 \( 1 + (319. + 552. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (1.29e3 - 747. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.43e3T + 7.04e5T^{2} \)
97 \( 1 + (-772. + 445. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.84947558129285160661266392677, −11.34347603495678443395673052799, −10.24094732253716939696261347016, −9.017925671940999482592082942656, −7.46596871827909278893297577886, −6.48758588789474713007791320074, −5.19628422728038862574161751600, −4.54107532225363014044598852534, −3.06065652772854343428097976860, −1.40315306137830026438656330660, 0.866919529283465979061566506181, 3.58759277903379197853129192205, 4.44409740452467061348113582600, 5.85230185379271667943717966573, 6.06400991165962727585600296819, 7.34039300086467745275680630511, 8.828232663800438998542390285094, 10.10899325707767786414206528006, 10.90528919722126044787820091410, 11.99686760387290376317916304522

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