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Properties

Label 2-15e2-9.4-c3-0-33
Degree $2$
Conductor $225$
Sign $0.999 + 0.0230i$
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  + (1.52 + 2.63i)2-s + (−4.01 + 3.29i)3-s + (−0.626 + 1.08i)4-s + (−14.7 − 5.57i)6-s + (−6.85 − 11.8i)7-s + 20.5·8-s + (5.29 − 26.4i)9-s + (−15.9 − 27.5i)11-s + (−1.05 − 6.42i)12-s + (29.1 − 50.4i)13-s + (20.8 − 36.1i)14-s + (36.2 + 62.7i)16-s − 109.·17-s + (77.8 − 26.3i)18-s + 129.·19-s + ⋯
L(s)  = 1  + (0.537 + 0.931i)2-s + (−0.773 + 0.633i)3-s + (−0.0782 + 0.135i)4-s + (−1.00 − 0.379i)6-s + (−0.370 − 0.641i)7-s + 0.907·8-s + (0.196 − 0.980i)9-s + (−0.435 − 0.755i)11-s + (−0.0254 − 0.154i)12-s + (0.621 − 1.07i)13-s + (0.398 − 0.689i)14-s + (0.566 + 0.980i)16-s − 1.55·17-s + (1.01 − 0.344i)18-s + 1.56·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0230i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.999 + 0.0230i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.66476 - 0.0191651i\)
\(L(\frac12)\) \(\approx\) \(1.66476 - 0.0191651i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.01 - 3.29i)T \)
5 \( 1 \)
good2 \( 1 + (-1.52 - 2.63i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (6.85 + 11.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (15.9 + 27.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-29.1 + 50.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 109.T + 4.91e3T^{2} \)
19 \( 1 - 129.T + 6.85e3T^{2} \)
23 \( 1 + (-39.8 + 68.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (4.51 + 7.82i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-16.6 + 28.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 22.1T + 5.06e4T^{2} \)
41 \( 1 + (60.8 - 105. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-5.07 - 8.78i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (220. + 382. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 593.T + 1.48e5T^{2} \)
59 \( 1 + (221. - 383. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (72.2 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-431. + 747. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 818.T + 3.57e5T^{2} \)
73 \( 1 + 495.T + 3.89e5T^{2} \)
79 \( 1 + (585. + 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (212. + 367. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + (-799. - 1.38e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.62913125542613259219229323242, −10.72166842808960378582558899284, −10.16934570323290334363721820589, −8.739450230602922389350702361696, −7.41761377638814783332514379340, −6.43901580094562594326107397116, −5.61606900646438776958026327871, −4.68342677375771006346883933583, −3.43166526078653868347345411701, −0.67028637445166750810915321413, 1.54214320749757017199127086241, 2.66579942806813224169788850124, 4.29195469619074718861931570376, 5.36366651459053861139901287248, 6.69278158418459212484212302650, 7.53619605695109408284815498562, 9.031564229229597542171823527150, 10.22174357354167609052981521606, 11.37897183093743754671742038406, 11.65839552991584651481272912126

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