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Properties

Label 2-15e2-45.4-c3-0-2
Degree $2$
Conductor $225$
Sign $-0.772 - 0.634i$
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  + (−4.66 + 2.69i)2-s + (3.19 − 4.09i)3-s + (10.4 − 18.1i)4-s + (−3.88 + 27.7i)6-s + (−10.8 + 6.28i)7-s + 69.9i·8-s + (−6.54 − 26.1i)9-s + (−6.41 − 11.1i)11-s + (−40.9 − 101. i)12-s + (−51.8 − 29.9i)13-s + (33.8 − 58.6i)14-s + (−104. − 180. i)16-s + 110. i·17-s + (101. + 104. i)18-s + 12.0·19-s + ⋯
L(s)  = 1  + (−1.64 + 0.951i)2-s + (0.615 − 0.788i)3-s + (1.31 − 2.27i)4-s + (−0.264 + 1.88i)6-s + (−0.587 + 0.339i)7-s + 3.09i·8-s + (−0.242 − 0.970i)9-s + (−0.175 − 0.304i)11-s + (−0.983 − 2.43i)12-s + (−1.10 − 0.638i)13-s + (0.646 − 1.11i)14-s + (−1.63 − 2.82i)16-s + 1.56i·17-s + (1.32 + 1.36i)18-s + 0.145·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.772 - 0.634i$
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: \((2,\ 225,\ (\ :3/2),\ -0.772 - 0.634i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.106766 + 0.298165i\)
\(L(\frac12)\) \(\approx\) \(0.106766 + 0.298165i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.19 + 4.09i)T \)
5 \( 1 \)
good2 \( 1 + (4.66 - 2.69i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (10.8 - 6.28i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (6.41 + 11.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (51.8 + 29.9i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 110. iT - 4.91e3T^{2} \)
19 \( 1 - 12.0T + 6.85e3T^{2} \)
23 \( 1 + (-58.6 - 33.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-99.9 - 173. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (38.3 - 66.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 22.4iT - 5.06e4T^{2} \)
41 \( 1 + (43.8 - 76.0i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-103. + 59.7i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (210. - 121. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 293. iT - 1.48e5T^{2} \)
59 \( 1 + (290. - 503. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-386. - 670. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (200. + 115. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 744.T + 3.57e5T^{2} \)
73 \( 1 + 264. iT - 3.89e5T^{2} \)
79 \( 1 + (-279. - 484. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (1.05e3 - 610. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 255.T + 7.04e5T^{2} \)
97 \( 1 + (908. - 524. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.15122221963536783441128650396, −10.74315694352325384692145499797, −9.873073768408161282917406688052, −8.950971329478392583530868638576, −8.239169454738887243249946107094, −7.37403481590742876478581019336, −6.51885463238004630704861826394, −5.56186854206925424026187670456, −2.82795180744431176784052797484, −1.35134303907061797539826195026, 0.21041077370654634928760435149, 2.25917433035915036224807265262, 3.16116411316589866795587268126, 4.58704112007364875165667291908, 6.98010004835173048547472465811, 7.73611414033050423921362966740, 8.884934854866416241996664924498, 9.707521181716674866489179910093, 9.979725422023177959728494540837, 11.14265373199605804067693445274

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