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Properties

Label 2-15e2-9.7-c3-0-28
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: $\chi_{225} (151, \cdot )$
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

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

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