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Properties

Label 2-15e2-15.8-c3-0-8
Degree $2$
Conductor $225$
Sign $0.0618 - 0.998i$
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  + (2.12 + 2.12i)2-s + 0.999i·4-s + (−9 + 9i)7-s + (14.8 − 14.8i)8-s + 38.1i·11-s + (63 + 63i)13-s − 38.1·14-s + 71·16-s + (29.6 + 29.6i)17-s + 70i·19-s + (−81 + 81i)22-s + (72.1 − 72.1i)23-s + 267. i·26-s + (−8.99 − 8.99i)28-s − 229.·29-s + ⋯
L(s)  = 1  + (0.749 + 0.749i)2-s + 0.124i·4-s + (−0.485 + 0.485i)7-s + (0.656 − 0.656i)8-s + 1.04i·11-s + (1.34 + 1.34i)13-s − 0.728·14-s + 1.10·16-s + (0.423 + 0.423i)17-s + 0.845i·19-s + (−0.784 + 0.784i)22-s + (0.653 − 0.653i)23-s + 2.01i·26-s + (−0.0607 − 0.0607i)28-s − 1.46·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.0618 - 0.998i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.0618 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.93057 + 1.81461i\)
\(L(\frac12)\) \(\approx\) \(1.93057 + 1.81461i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-2.12 - 2.12i)T + 8iT^{2} \)
7 \( 1 + (9 - 9i)T - 343iT^{2} \)
11 \( 1 - 38.1iT - 1.33e3T^{2} \)
13 \( 1 + (-63 - 63i)T + 2.19e3iT^{2} \)
17 \( 1 + (-29.6 - 29.6i)T + 4.91e3iT^{2} \)
19 \( 1 - 70iT - 6.85e3T^{2} \)
23 \( 1 + (-72.1 + 72.1i)T - 1.21e4iT^{2} \)
29 \( 1 + 229.T + 2.43e4T^{2} \)
31 \( 1 - 196T + 2.97e4T^{2} \)
37 \( 1 + (-207 + 207i)T - 5.06e4iT^{2} \)
41 \( 1 - 267. iT - 6.89e4T^{2} \)
43 \( 1 + (144 + 144i)T + 7.95e4iT^{2} \)
47 \( 1 + (356. + 356. i)T + 1.03e5iT^{2} \)
53 \( 1 + (224. - 224. i)T - 1.48e5iT^{2} \)
59 \( 1 + 267.T + 2.05e5T^{2} \)
61 \( 1 + 322T + 2.26e5T^{2} \)
67 \( 1 + (-378 + 378i)T - 3.00e5iT^{2} \)
71 \( 1 + 840. iT - 3.57e5T^{2} \)
73 \( 1 + (-378 - 378i)T + 3.89e5iT^{2} \)
79 \( 1 - 488iT - 4.93e5T^{2} \)
83 \( 1 + (-772. + 772. i)T - 5.71e5iT^{2} \)
89 \( 1 + 267.T + 7.04e5T^{2} \)
97 \( 1 + (252 - 252i)T - 9.12e5iT^{2} \)
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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

−12.34318794769150897373086856561, −11.14882423179248492001429278758, −9.991369552569039143728341578901, −9.115998302257311390789916180716, −7.78457504866236935251638874729, −6.58995680535944119871169699128, −6.02759790741328477042840901861, −4.71668649373565350132669350449, −3.69765596098251923377977862788, −1.66612519306822245890503471034, 0.967239817515722642330527394836, 3.01883206035402986110805243031, 3.59585113682796791094767826017, 5.07546082215536363680178892796, 6.16614641491581033910791772439, 7.63988272991155122013862636542, 8.545552111989571715238918445272, 9.904997173239584814307937341986, 11.06150234015556429580939962494, 11.35141726013058466014390186558

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