Properties

Degree $2$
Conductor $360$
Sign $-0.447 - 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + i)5-s + 2i·7-s − 2·11-s + 2i·13-s + 6i·17-s − 8·19-s + 4i·23-s + (3 − 4i)25-s + 8·29-s + (−2 − 4i)35-s + 10i·37-s − 2·41-s − 12i·43-s + 3·49-s − 10i·53-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s + 0.755i·7-s − 0.603·11-s + 0.554i·13-s + 1.45i·17-s − 1.83·19-s + 0.834i·23-s + (0.600 − 0.800i)25-s + 1.48·29-s + (−0.338 − 0.676i)35-s + 1.64i·37-s − 0.312·41-s − 1.82i·43-s + 0.428·49-s − 1.37i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.447 - 0.894i$
Motivic weight: \(1\)
Character: $\chi_{360} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.414048 + 0.669943i\)
\(L(\frac12)\) \(\approx\) \(0.414048 + 0.669943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2 - i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−11.80210888241706262621298027878, −10.80040561276049023925394338788, −10.13132762261123793513291037470, −8.585616972306893103939095061379, −8.281361488778413926463002734989, −6.93043854564591477779557992318, −6.08256025102826789712467462417, −4.70015757120587292748410980883, −3.60070950372770491348534872187, −2.21230070927819415853594571671, 0.52628682333098433130296392786, 2.77212848526543163278910110492, 4.19791095133386478732424228648, 4.94091759316762413426731784931, 6.45518930545176223838856956012, 7.51141579759640834159242484090, 8.217447225720674731267822909093, 9.200996140699861748593738318899, 10.49328321214195567745791798463, 10.97047280000368757684386671512

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