Properties

Label 2-165-33.32-c5-0-15
Degree $2$
Conductor $165$
Sign $0.595 - 0.803i$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.66·2-s + (8.51 − 13.0i)3-s − 18.5·4-s + 25i·5-s + (−31.1 + 47.8i)6-s − 5.76i·7-s + 185.·8-s + (−97.9 − 222. i)9-s − 91.5i·10-s + (−23.8 + 400. i)11-s + (−158. + 242. i)12-s − 1.00e3i·13-s + 21.1i·14-s + (326. + 212. i)15-s − 83.4·16-s − 2.08e3·17-s + ⋯
L(s)  = 1  − 0.647·2-s + (0.546 − 0.837i)3-s − 0.580·4-s + 0.447i·5-s + (−0.353 + 0.542i)6-s − 0.0445i·7-s + 1.02·8-s + (−0.402 − 0.915i)9-s − 0.289i·10-s + (−0.0594 + 0.998i)11-s + (−0.317 + 0.486i)12-s − 1.64i·13-s + 0.0288i·14-s + (0.374 + 0.244i)15-s − 0.0814·16-s − 1.74·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.595 - 0.803i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8802117321\)
\(L(\frac12)\) \(\approx\) \(0.8802117321\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.51 + 13.0i)T \)
5 \( 1 - 25iT \)
11 \( 1 + (23.8 - 400. i)T \)
good2 \( 1 + 3.66T + 32T^{2} \)
7 \( 1 + 5.76iT - 1.68e4T^{2} \)
13 \( 1 + 1.00e3iT - 3.71e5T^{2} \)
17 \( 1 + 2.08e3T + 1.41e6T^{2} \)
19 \( 1 - 1.30e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.66e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.16e3T + 2.05e7T^{2} \)
31 \( 1 - 2.35e3T + 2.86e7T^{2} \)
37 \( 1 - 6.46e3T + 6.93e7T^{2} \)
41 \( 1 - 1.89e4T + 1.15e8T^{2} \)
43 \( 1 + 6.82e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.17e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.92e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.97e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.26e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.99e4T + 1.35e9T^{2} \)
71 \( 1 - 1.30e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.97e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.22e4iT - 3.07e9T^{2} \)
83 \( 1 - 3.73e4T + 3.93e9T^{2} \)
89 \( 1 - 1.55e4iT - 5.58e9T^{2} \)
97 \( 1 + 7.17e4T + 8.58e9T^{2} \)
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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

−12.27110336847410480714025490019, −10.85878261918562745592637016218, −9.871965686077292475771765466202, −8.953250492264385858474778466079, −7.85894074064072765401505146513, −7.29647495936235742029548988134, −5.80710162166794449684307389726, −4.14117740877172834581336187590, −2.57897058252078543137695997980, −1.11639085753871342938646578952, 0.39602911883300307498763427285, 2.33476258204787434551993492967, 4.20785164245998338318434870375, 4.74230818885261141892241934442, 6.55419829889644464491202179531, 8.208965540936210640714802831277, 8.893548576517509514213008073105, 9.356657034787955413388697999563, 10.63802392694927647409854028749, 11.37893733925881405376467031954

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