Properties

Label 2-15e2-9.4-c3-0-47
Degree $2$
Conductor $225$
Sign $-0.890 + 0.454i$
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  + (−0.238 − 0.412i)2-s + (3.09 − 4.17i)3-s + (3.88 − 6.73i)4-s + (−2.45 − 0.279i)6-s + (−6.34 − 10.9i)7-s − 7.50·8-s + (−7.90 − 25.8i)9-s + (0.794 + 1.37i)11-s + (−16.1 − 37.0i)12-s + (−5.36 + 9.29i)13-s + (−3.01 + 5.22i)14-s + (−29.3 − 50.7i)16-s − 69.7·17-s + (−8.76 + 9.40i)18-s + 98.5·19-s + ⋯
L(s)  = 1  + (−0.0841 − 0.145i)2-s + (0.594 − 0.803i)3-s + (0.485 − 0.841i)4-s + (−0.167 − 0.0190i)6-s + (−0.342 − 0.592i)7-s − 0.331·8-s + (−0.292 − 0.956i)9-s + (0.0217 + 0.0377i)11-s + (−0.387 − 0.891i)12-s + (−0.114 + 0.198i)13-s + (−0.0576 + 0.0997i)14-s + (−0.457 − 0.793i)16-s − 0.995·17-s + (−0.114 + 0.123i)18-s + 1.19·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.890 + 0.454i$
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: \((2,\ 225,\ (\ :3/2),\ -0.890 + 0.454i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.435886 - 1.81431i\)
\(L(\frac12)\) \(\approx\) \(0.435886 - 1.81431i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.09 + 4.17i)T \)
5 \( 1 \)
good2 \( 1 + (0.238 + 0.412i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (6.34 + 10.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-0.794 - 1.37i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (5.36 - 9.29i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 69.7T + 4.91e3T^{2} \)
19 \( 1 - 98.5T + 6.85e3T^{2} \)
23 \( 1 + (-15.7 + 27.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-150. - 260. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-58.6 + 101. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 169.T + 5.06e4T^{2} \)
41 \( 1 + (70.9 - 122. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (150. + 260. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (243. + 422. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 459.T + 1.48e5T^{2} \)
59 \( 1 + (-250. + 433. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (290. + 503. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (250. - 433. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 - 435.T + 3.89e5T^{2} \)
79 \( 1 + (187. + 325. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-646. - 1.11e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 403.T + 7.04e5T^{2} \)
97 \( 1 + (-790. - 1.36e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.44533608568164318533412078027, −10.36681264562323196450566279426, −9.469251370983235058488764344327, −8.450494335661164080006451008817, −7.01895906931492066732095734426, −6.66665150179872637637584357475, −5.17876690288073989308095506432, −3.38698185216814078747576105564, −2.04305789009412974959206861849, −0.71255198024751090257940048072, 2.44439796770123840795325216447, 3.35297185462998795710358352621, 4.66477349966933203795601874461, 6.10266863169093142663070245789, 7.39218313435238997694450438905, 8.368523471857071646979196674393, 9.146465443908598191321060031061, 10.13925454627817114593519499179, 11.31184011250332338400675683610, 12.07529086169086914737189877330

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