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Properties

Label 2-360-5.4-c3-0-11
Degree $2$
Conductor $360$
Sign $0.983 - 0.178i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
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  + (2 + 11i)5-s − 10i·7-s + 14·11-s − 82i·13-s + 18i·17-s + 136·19-s + 140i·23-s + (−117 + 44i)25-s + 112·29-s + 72·31-s + (110 − 20i)35-s − 26i·37-s + 446·41-s + 396i·43-s − 144i·47-s + ⋯
L(s)  = 1  + (0.178 + 0.983i)5-s − 0.539i·7-s + 0.383·11-s − 1.74i·13-s + 0.256i·17-s + 1.64·19-s + 1.26i·23-s + (−0.936 + 0.351i)25-s + 0.717·29-s + 0.417·31-s + (0.531 − 0.0965i)35-s − 0.115i·37-s + 1.69·41-s + 1.40i·43-s − 0.446i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.983 - 0.178i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ 0.983 - 0.178i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.044775669\)
\(L(\frac12)\) \(\approx\) \(2.044775669\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2 - 11i)T \)
good7 \( 1 + 10iT - 343T^{2} \)
11 \( 1 - 14T + 1.33e3T^{2} \)
13 \( 1 + 82iT - 2.19e3T^{2} \)
17 \( 1 - 18iT - 4.91e3T^{2} \)
19 \( 1 - 136T + 6.85e3T^{2} \)
23 \( 1 - 140iT - 1.21e4T^{2} \)
29 \( 1 - 112T + 2.43e4T^{2} \)
31 \( 1 - 72T + 2.97e4T^{2} \)
37 \( 1 + 26iT - 5.06e4T^{2} \)
41 \( 1 - 446T + 6.89e4T^{2} \)
43 \( 1 - 396iT - 7.95e4T^{2} \)
47 \( 1 + 144iT - 1.03e5T^{2} \)
53 \( 1 + 158iT - 1.48e5T^{2} \)
59 \( 1 + 342T + 2.05e5T^{2} \)
61 \( 1 - 314T + 2.26e5T^{2} \)
67 \( 1 - 152iT - 3.00e5T^{2} \)
71 \( 1 - 932T + 3.57e5T^{2} \)
73 \( 1 + 548iT - 3.89e5T^{2} \)
79 \( 1 - 512T + 4.93e5T^{2} \)
83 \( 1 + 284iT - 5.71e5T^{2} \)
89 \( 1 + 810T + 7.04e5T^{2} \)
97 \( 1 + 1.30e3iT - 9.12e5T^{2} \)
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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

−10.95942559590852350168536932705, −10.16015032991410822152717273443, −9.465120887486258200383613860814, −7.920448015098797630482026436760, −7.39610660430239996049992416897, −6.21316289947987616472946346655, −5.28579352766365575201061746859, −3.68459536739833001781898964659, −2.83646862130875683259825261545, −1.00607236108741741676594706908, 1.01255534666092581603707822979, 2.37945471464623278318545644253, 4.09305679195953190857447899114, 4.99896818040454582588501567805, 6.09587686307935641331794205028, 7.15199360836763266471970255173, 8.423377856454973128598194351028, 9.167631996572965760356471178786, 9.755648475468319627177722749736, 11.18917586318185297184829883538

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