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Properties

Label 2-15e2-45.34-c3-0-33
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: $\chi_{225} (124, \cdot )$
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

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

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