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Properties

Label 2-15e2-9.4-c3-0-32
Degree $2$
Conductor $225$
Sign $0.998 - 0.0478i$
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  + (0.785 + 1.35i)2-s + (3.89 − 3.43i)3-s + (2.76 − 4.79i)4-s + (7.73 + 2.60i)6-s + (17.1 + 29.6i)7-s + 21.2·8-s + (3.40 − 26.7i)9-s + (13.4 + 23.2i)11-s + (−5.67 − 28.1i)12-s + (−9.74 + 16.8i)13-s + (−26.8 + 46.5i)14-s + (−5.45 − 9.44i)16-s − 29.1·17-s + (39.0 − 16.3i)18-s + 47.1·19-s + ⋯
L(s)  = 1  + (0.277 + 0.480i)2-s + (0.750 − 0.660i)3-s + (0.345 − 0.599i)4-s + (0.526 + 0.177i)6-s + (0.924 + 1.60i)7-s + 0.939·8-s + (0.126 − 0.991i)9-s + (0.368 + 0.638i)11-s + (−0.136 − 0.678i)12-s + (−0.207 + 0.360i)13-s + (−0.513 + 0.888i)14-s + (−0.0852 − 0.147i)16-s − 0.415·17-s + (0.511 − 0.214i)18-s + 0.568·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0478i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.998 - 0.0478i$
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: \((2,\ 225,\ (\ :3/2),\ 0.998 - 0.0478i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.28794 + 0.0787768i\)
\(L(\frac12)\) \(\approx\) \(3.28794 + 0.0787768i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.89 + 3.43i)T \)
5 \( 1 \)
good2 \( 1 + (-0.785 - 1.35i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-17.1 - 29.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-13.4 - 23.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (9.74 - 16.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 29.1T + 4.91e3T^{2} \)
19 \( 1 - 47.1T + 6.85e3T^{2} \)
23 \( 1 + (-56.6 + 98.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (40.6 + 70.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-5.41 + 9.38i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 410.T + 5.06e4T^{2} \)
41 \( 1 + (-221. + 384. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-169. - 294. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-118. - 204. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 609.T + 1.48e5T^{2} \)
59 \( 1 + (7.77 - 13.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-108. + 187. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 65.4T + 3.57e5T^{2} \)
73 \( 1 + 711.T + 3.89e5T^{2} \)
79 \( 1 + (-478. - 829. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (261. + 452. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + (-400. - 694. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.98796820158666700618000495224, −11.03256020412045143659095019363, −9.531197609053629597767355032158, −8.783716638882080549746376658393, −7.69640267298027781453682334470, −6.73267782564197022212320240880, −5.70623821404216634747361116770, −4.58731308247031225013397652369, −2.50070813824748647952774964365, −1.61293851890339520914912192910, 1.51868884519345636625275681799, 3.18938249886562503496248048635, 3.99327548037635216679437267648, 5.00053962345010567023318146110, 7.14715207982555574950266412101, 7.76945786967126016929932370411, 8.773923131996189035860688882175, 10.12778860511443814640917835253, 10.91804821882763496505548122430, 11.47147980750435503719091438185

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