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Properties

Label 2-15e2-45.4-c3-0-7
Degree $2$
Conductor $225$
Sign $0.125 - 0.992i$
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  + (−3.96 + 2.28i)2-s + (−3.96 − 3.36i)3-s + (6.45 − 11.1i)4-s + (23.3 + 4.26i)6-s + (−17.4 + 10.0i)7-s + 22.4i·8-s + (4.37 + 26.6i)9-s + (−33.1 − 57.4i)11-s + (−63.2 + 22.5i)12-s + (−40.5 − 23.4i)13-s + (45.9 − 79.6i)14-s + (0.237 + 0.411i)16-s − 47.6i·17-s + (−78.2 − 95.5i)18-s + 9.95·19-s + ⋯
L(s)  = 1  + (−1.40 + 0.808i)2-s + (−0.762 − 0.647i)3-s + (0.807 − 1.39i)4-s + (1.59 + 0.290i)6-s + (−0.940 + 0.543i)7-s + 0.993i·8-s + (0.162 + 0.986i)9-s + (−0.909 − 1.57i)11-s + (−1.52 + 0.543i)12-s + (−0.864 − 0.499i)13-s + (0.878 − 1.52i)14-s + (0.00371 + 0.00643i)16-s − 0.679i·17-s + (−1.02 − 1.25i)18-s + 0.120·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.125 - 0.992i$
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: \((2,\ 225,\ (\ :3/2),\ 0.125 - 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.162177 + 0.142886i\)
\(L(\frac12)\) \(\approx\) \(0.162177 + 0.142886i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.96 + 3.36i)T \)
5 \( 1 \)
good2 \( 1 + (3.96 - 2.28i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (17.4 - 10.0i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (33.1 + 57.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (40.5 + 23.4i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 47.6iT - 4.91e3T^{2} \)
19 \( 1 - 9.95T + 6.85e3T^{2} \)
23 \( 1 + (-8.30 - 4.79i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (89.3 + 154. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (77.0 - 133. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 248. iT - 5.06e4T^{2} \)
41 \( 1 + (124. - 216. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-183. + 106. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-411. + 237. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 546. iT - 1.48e5T^{2} \)
59 \( 1 + (209. - 363. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-272. - 472. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-387. - 223. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 409.T + 3.57e5T^{2} \)
73 \( 1 - 358. iT - 3.89e5T^{2} \)
79 \( 1 + (325. + 564. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-704. + 406. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 200.T + 7.04e5T^{2} \)
97 \( 1 + (-218. + 126. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.85017170856885657632942475318, −10.76398250379792758028339592083, −9.991213706103682044499490843676, −8.916394935969489158371538877264, −7.948483354543834594902742084311, −7.14068540844589830965142157380, −6.07179508261047286516435117047, −5.43339106501066683629015570736, −2.74962982987454283200524001505, −0.69054609298924393223449848816, 0.25724540645996849083565586677, 2.14226380213553347537707675049, 3.76491856448478235070325835584, 5.16554351292150656769542992383, 6.83994067985770868660569574804, 7.62892558380576953856525119456, 9.243587489121576350665152274067, 9.748576934122751381069200350890, 10.42758868830739448887944420333, 11.13963221416478012480958422071

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