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Properties

Label 2-1620-20.19-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.866 - 0.5i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + (−0.866 − 0.5i)5-s − i·8-s + (0.5 − 0.866i)10-s + 1.73i·13-s + 16-s + i·17-s + (0.866 + 0.5i)20-s + (0.499 + 0.866i)25-s − 1.73·26-s − 1.73·29-s + i·32-s − 34-s + 1.73i·37-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.866 − 0.5i)5-s − i·8-s + (0.5 − 0.866i)10-s + 1.73i·13-s + 16-s + i·17-s + (0.866 + 0.5i)20-s + (0.499 + 0.866i)25-s − 1.73·26-s − 1.73·29-s + i·32-s − 34-s + 1.73i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6301061146\)
\(L(\frac12)\) \(\approx\) \(0.6301061146\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.483095798082739543834690162874, −9.015571638429939169397780475214, −8.229088517772982115398264629471, −7.55818381929623505522924654034, −6.75944376907471394063460209038, −6.01228807099853720217802480619, −4.91390325605339482359723327636, −4.24983782223949627402055525980, −3.51821140223809169804125484424, −1.56948359724535365535432807628, 0.51635718238333173719613454758, 2.28736253099465935393648304193, 3.25285946103130417506067223746, 3.84580058384735975047970854514, 5.01976373326022818927485637194, 5.70520925075173356092618473702, 7.10909113420385988617202901833, 7.81386143180554412405658650264, 8.469898183585016074267851824810, 9.469566058352512460194360437554

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