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Properties

Label 2-15e2-9.7-c3-0-36
Degree $2$
Conductor $225$
Sign $0.982 + 0.185i$
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  + (−0.392 + 0.680i)2-s + (−3.69 + 3.65i)3-s + (3.69 + 6.39i)4-s + (−1.03 − 3.94i)6-s + (10.4 − 18.1i)7-s − 12.0·8-s + (0.319 − 26.9i)9-s + (29.6 − 51.4i)11-s + (−36.9 − 10.1i)12-s + (−22.9 − 39.7i)13-s + (8.21 + 14.2i)14-s + (−24.7 + 42.9i)16-s − 43.0·17-s + (18.2 + 10.8i)18-s + 140.·19-s + ⋯
L(s)  = 1  + (−0.138 + 0.240i)2-s + (−0.711 + 0.702i)3-s + (0.461 + 0.799i)4-s + (−0.0702 − 0.268i)6-s + (0.564 − 0.977i)7-s − 0.533·8-s + (0.0118 − 0.999i)9-s + (0.813 − 1.40i)11-s + (−0.890 − 0.244i)12-s + (−0.489 − 0.848i)13-s + (0.156 + 0.271i)14-s + (−0.387 + 0.670i)16-s − 0.614·17-s + (0.238 + 0.141i)18-s + 1.69·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.982 + 0.185i$
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: \((2,\ 225,\ (\ :3/2),\ 0.982 + 0.185i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.34081 - 0.125308i\)
\(L(\frac12)\) \(\approx\) \(1.34081 - 0.125308i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.69 - 3.65i)T \)
5 \( 1 \)
good2 \( 1 + (0.392 - 0.680i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-10.4 + 18.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-29.6 + 51.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (22.9 + 39.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 43.0T + 4.91e3T^{2} \)
19 \( 1 - 140.T + 6.85e3T^{2} \)
23 \( 1 + (50.8 + 88.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (6.19 - 10.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (35.9 + 62.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 150.T + 5.06e4T^{2} \)
41 \( 1 + (25.5 + 44.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (17.2 - 29.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-58.9 + 102. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 137.T + 1.48e5T^{2} \)
59 \( 1 + (248. + 429. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-123. + 214. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-413. - 715. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 260.T + 3.57e5T^{2} \)
73 \( 1 - 372.T + 3.89e5T^{2} \)
79 \( 1 + (238. - 412. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-156. + 271. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 817.T + 7.04e5T^{2} \)
97 \( 1 + (-470. + 815. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.45891670370038623297109691592, −11.08180478548973148562526665516, −9.936052653130396655250292076588, −8.748526211736000607369611777230, −7.72464527579264095080395361377, −6.68727026988365335876049019203, −5.59190195897914793731481730936, −4.19030815254810542195588883510, −3.20343001498158524675406702543, −0.68159319427352616619949085142, 1.43571505565665016997387872974, 2.25711383442695895625491728612, 4.74192519090869358238627252353, 5.64539884536606041286833601685, 6.73453891973906984855212176611, 7.52222504400995408662207423422, 9.159451783687104228220573941541, 9.866817795302165210089861462566, 11.19242646103060366034206033533, 11.87842164611412511855424984819

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