Properties

Label 2-15e2-9.4-c3-0-36
Degree $2$
Conductor $225$
Sign $0.998 + 0.0465i$
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  + (1.87 + 3.24i)2-s + (4.05 − 3.24i)3-s + (−3.02 + 5.24i)4-s + (18.1 + 7.08i)6-s + (−15.6 − 27.1i)7-s + 7.30·8-s + (5.92 − 26.3i)9-s + (10.4 + 18.0i)11-s + (4.73 + 31.0i)12-s + (29.9 − 51.9i)13-s + (58.7 − 101. i)14-s + (37.8 + 65.6i)16-s + 74.0·17-s + (96.6 − 30.1i)18-s − 63.8·19-s + ⋯
L(s)  = 1  + (0.662 + 1.14i)2-s + (0.780 − 0.624i)3-s + (−0.378 + 0.655i)4-s + (1.23 + 0.482i)6-s + (−0.846 − 1.46i)7-s + 0.322·8-s + (0.219 − 0.975i)9-s + (0.285 + 0.494i)11-s + (0.113 + 0.747i)12-s + (0.639 − 1.10i)13-s + (1.12 − 1.94i)14-s + (0.592 + 1.02i)16-s + 1.05·17-s + (1.26 − 0.394i)18-s − 0.770·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0465i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.998 + 0.0465i$
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: \((2,\ 225,\ (\ :3/2),\ 0.998 + 0.0465i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.22144 - 0.0750941i\)
\(L(\frac12)\) \(\approx\) \(3.22144 - 0.0750941i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.05 + 3.24i)T \)
5 \( 1 \)
good2 \( 1 + (-1.87 - 3.24i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (15.6 + 27.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-10.4 - 18.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-29.9 + 51.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 74.0T + 4.91e3T^{2} \)
19 \( 1 + 63.8T + 6.85e3T^{2} \)
23 \( 1 + (16.4 - 28.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-80.0 - 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-127. + 220. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 215.T + 5.06e4T^{2} \)
41 \( 1 + (70.8 - 122. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-68.9 - 119. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-16.7 - 29.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 41.9T + 1.48e5T^{2} \)
59 \( 1 + (307. - 532. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-67.1 - 116. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (428. - 742. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 588.T + 3.57e5T^{2} \)
73 \( 1 - 618.T + 3.89e5T^{2} \)
79 \( 1 + (-172. - 299. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (546. + 946. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 414.T + 7.04e5T^{2} \)
97 \( 1 + (100. + 174. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.37100784296103442104892366618, −10.59412708660090907107826263904, −9.827278011151568381953270520736, −8.312170957372410036804466861831, −7.51659795008096404373493092731, −6.79775036130904985070065340686, −5.91471538924932231115393347186, −4.25480785416592428454682115674, −3.31598660211667617841969304349, −1.09531839906663336071761038622, 1.94709663058904067023465157120, 3.01648227500858694782180936915, 3.84862751613194386520163559896, 5.13829287950008029474942067056, 6.45394321686032302094224276180, 8.265008754714267883638295407595, 9.083982141253408687652312425111, 9.957258311301065511291216135618, 10.91490184077712525176006708252, 12.00454121296696012044237373026

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