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Properties

Label 2-15e2-45.34-c3-0-22
Degree $2$
Conductor $225$
Sign $-0.0441 - 0.999i$
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  + (4.66 + 2.69i)2-s + (−3.19 − 4.09i)3-s + (10.4 + 18.1i)4-s + (−3.88 − 27.7i)6-s + (10.8 + 6.28i)7-s + 69.9i·8-s + (−6.54 + 26.1i)9-s + (−6.41 + 11.1i)11-s + (40.9 − 101. i)12-s + (51.8 − 29.9i)13-s + (33.8 + 58.6i)14-s + (−104. + 180. i)16-s + 110. i·17-s + (−101. + 104. i)18-s + 12.0·19-s + ⋯
L(s)  = 1  + (1.64 + 0.951i)2-s + (−0.615 − 0.788i)3-s + (1.31 + 2.27i)4-s + (−0.264 − 1.88i)6-s + (0.587 + 0.339i)7-s + 3.09i·8-s + (−0.242 + 0.970i)9-s + (−0.175 + 0.304i)11-s + (0.983 − 2.43i)12-s + (1.10 − 0.638i)13-s + (0.646 + 1.11i)14-s + (−1.63 + 2.82i)16-s + 1.56i·17-s + (−1.32 + 1.36i)18-s + 0.145·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0441 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0441 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.0441 - 0.999i$
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: \((2,\ 225,\ (\ :3/2),\ -0.0441 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.70559 + 2.82791i\)
\(L(\frac12)\) \(\approx\) \(2.70559 + 2.82791i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.19 + 4.09i)T \)
5 \( 1 \)
good2 \( 1 + (-4.66 - 2.69i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (-10.8 - 6.28i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (6.41 - 11.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-51.8 + 29.9i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 110. iT - 4.91e3T^{2} \)
19 \( 1 - 12.0T + 6.85e3T^{2} \)
23 \( 1 + (58.6 - 33.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-99.9 + 173. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (38.3 + 66.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 22.4iT - 5.06e4T^{2} \)
41 \( 1 + (43.8 + 76.0i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (103. + 59.7i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-210. - 121. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 293. iT - 1.48e5T^{2} \)
59 \( 1 + (290. + 503. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-386. + 670. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-200. + 115. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 744.T + 3.57e5T^{2} \)
73 \( 1 + 264. iT - 3.89e5T^{2} \)
79 \( 1 + (-279. + 484. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-1.05e3 - 610. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 255.T + 7.04e5T^{2} \)
97 \( 1 + (-908. - 524. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.34797271533952836262151248123, −11.57924457187387977933601722330, −10.67182796372305389318488737612, −8.247937002374165728587050158715, −7.87015137346418844391097450946, −6.50904009509035039719406188485, −5.90037222881453551584392068003, −4.99599659841418271992432997891, −3.71280751931013118822700388947, −2.03355772655765158931631844952, 1.14085396621587657357142018614, 2.98960545141889018541685446747, 4.11977624542888155208780197943, 4.91551708253592536422679937685, 5.84388330820345723433673553486, 6.91717894264332111146164120862, 9.009501794261492204860789278487, 10.22328029601199262998722188470, 10.94383361509448362261835800187, 11.57888963637023727383804242535

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