Properties

Label 2-15e2-25.16-c3-0-35
Degree $2$
Conductor $225$
Sign $-0.908 + 0.418i$
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  + (1.08 + 0.787i)2-s + (−1.91 − 5.90i)4-s + (3.83 − 10.5i)5-s − 12.2·7-s + (5.88 − 18.0i)8-s + (12.4 − 8.36i)10-s + (−2.00 − 1.45i)11-s + (−69.2 + 50.3i)13-s + (−13.2 − 9.61i)14-s + (−19.5 + 14.1i)16-s + (1.71 − 5.27i)17-s + (−7.45 + 22.9i)19-s + (−69.3 − 2.47i)20-s + (−1.02 − 3.16i)22-s + (−83.6 − 60.7i)23-s + ⋯
L(s)  = 1  + (0.383 + 0.278i)2-s + (−0.239 − 0.737i)4-s + (0.342 − 0.939i)5-s − 0.659·7-s + (0.259 − 0.799i)8-s + (0.392 − 0.264i)10-s + (−0.0550 − 0.0399i)11-s + (−1.47 + 1.07i)13-s + (−0.252 − 0.183i)14-s + (−0.305 + 0.221i)16-s + (0.0244 − 0.0752i)17-s + (−0.0900 + 0.277i)19-s + (−0.775 − 0.0276i)20-s + (−0.00995 − 0.0306i)22-s + (−0.758 − 0.550i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.908 + 0.418i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.197575 - 0.902010i\)
\(L(\frac12)\) \(\approx\) \(0.197575 - 0.902010i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.83 + 10.5i)T \)
good2 \( 1 + (-1.08 - 0.787i)T + (2.47 + 7.60i)T^{2} \)
7 \( 1 + 12.2T + 343T^{2} \)
11 \( 1 + (2.00 + 1.45i)T + (411. + 1.26e3i)T^{2} \)
13 \( 1 + (69.2 - 50.3i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-1.71 + 5.27i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (7.45 - 22.9i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + (83.6 + 60.7i)T + (3.75e3 + 1.15e4i)T^{2} \)
29 \( 1 + (-35.3 - 108. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-47.5 + 146. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-277. + 201. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (-28.0 + 20.3i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 154.T + 7.95e4T^{2} \)
47 \( 1 + (104. + 321. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (199. + 614. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-536. + 389. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-289. - 210. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + (164. - 505. i)T + (-2.43e5 - 1.76e5i)T^{2} \)
71 \( 1 + (-196. - 603. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (779. + 565. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-175. - 540. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-98.8 + 304. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + (1.07e3 + 784. i)T + (2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (-123. - 379. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−11.55868013488624062382000616906, −9.886531872123558892147995532870, −9.756030821494075212744111794787, −8.556875563182746467097226676362, −7.06531912319862631734159956173, −6.07857582088950379746589517486, −5.04296155640296357820258891408, −4.15775197847788724245535886672, −2.04493539823985232511454061049, −0.31520791500618934379341587588, 2.52916221994782244012203686361, 3.26351969175003447118674267473, 4.71389531183316385930775208756, 6.03726179558579119981536277698, 7.26820731700159343726627246837, 8.052936488561918697346323979417, 9.560640237658548256532870299467, 10.23619485917624639043575931379, 11.39779073341380704069077145458, 12.31112640909769852428647175044

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