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Properties

Label 2-15e2-45.34-c3-0-8
Degree $2$
Conductor $225$
Sign $-0.593 - 0.804i$
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  + (0.412 + 0.238i)2-s + (−4.17 + 3.09i)3-s + (−3.88 − 6.73i)4-s + (−2.45 + 0.279i)6-s + (10.9 + 6.34i)7-s − 7.50i·8-s + (7.90 − 25.8i)9-s + (0.794 − 1.37i)11-s + (37.0 + 16.1i)12-s + (9.29 − 5.36i)13-s + (3.01 + 5.22i)14-s + (−29.3 + 50.7i)16-s + 69.7i·17-s + (9.40 − 8.76i)18-s − 98.5·19-s + ⋯
L(s)  = 1  + (0.145 + 0.0841i)2-s + (−0.803 + 0.594i)3-s + (−0.485 − 0.841i)4-s + (−0.167 + 0.0190i)6-s + (0.592 + 0.342i)7-s − 0.331i·8-s + (0.292 − 0.956i)9-s + (0.0217 − 0.0377i)11-s + (0.891 + 0.387i)12-s + (0.198 − 0.114i)13-s + (0.0576 + 0.0997i)14-s + (−0.457 + 0.793i)16-s + 0.995i·17-s + (0.123 − 0.114i)18-s − 1.19·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.593 - 0.804i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.593 - 0.804i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.294176 + 0.582578i\)
\(L(\frac12)\) \(\approx\) \(0.294176 + 0.582578i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

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

−12.09633454675665013697742437594, −10.80959794566431613069960325510, −10.53302130643111385418849934335, −9.285783278158137889356456094128, −8.454310391920522558694012936047, −6.69202969110718273982258184297, −5.75659222635739464101290769060, −4.92375876381198129217290999521, −3.87071258630059824105617433987, −1.48137257831452671557915331076, 0.29813367247118921270286178167, 2.21717209218204824561287440899, 4.07809017990596754876168840290, 5.00738317085651071190602186509, 6.34982514107099941116856409045, 7.53444031082109918914908994940, 8.183824311039617894571585815476, 9.490058636697075353896313178199, 10.83895142795963620319542160110, 11.60272955992416084858782656178

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