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Properties

Label 2-360-120.77-c1-0-16
Degree $2$
Conductor $360$
Sign $0.826 + 0.563i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 − 0.319i)2-s + (1.79 − 0.879i)4-s + (−1.42 − 1.72i)5-s + (3.11 + 3.11i)7-s + (2.19 − 1.78i)8-s + (−2.51 − 1.91i)10-s + 1.17·11-s + (−2.15 − 2.15i)13-s + (5.29 + 3.30i)14-s + (2.45 − 3.15i)16-s + (1.33 − 1.33i)17-s − 0.322·19-s + (−4.07 − 1.84i)20-s + (1.62 − 0.375i)22-s + (−4.71 − 4.71i)23-s + ⋯
L(s)  = 1  + (0.974 − 0.225i)2-s + (0.898 − 0.439i)4-s + (−0.637 − 0.770i)5-s + (1.17 + 1.17i)7-s + (0.775 − 0.630i)8-s + (−0.795 − 0.606i)10-s + 0.355·11-s + (−0.596 − 0.596i)13-s + (1.41 + 0.882i)14-s + (0.613 − 0.789i)16-s + (0.323 − 0.323i)17-s − 0.0739·19-s + (−0.911 − 0.411i)20-s + (0.346 − 0.0801i)22-s + (−0.982 − 0.982i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.826 + 0.563i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.826 + 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29190 - 0.706692i\)
\(L(\frac12)\) \(\approx\) \(2.29190 - 0.706692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.319i)T \)
3 \( 1 \)
5 \( 1 + (1.42 + 1.72i)T \)
good7 \( 1 + (-3.11 - 3.11i)T + 7iT^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 + (2.15 + 2.15i)T + 13iT^{2} \)
17 \( 1 + (-1.33 + 1.33i)T - 17iT^{2} \)
19 \( 1 + 0.322T + 19T^{2} \)
23 \( 1 + (4.71 + 4.71i)T + 23iT^{2} \)
29 \( 1 - 6.63iT - 29T^{2} \)
31 \( 1 + 0.0675T + 31T^{2} \)
37 \( 1 + (7.60 - 7.60i)T - 37iT^{2} \)
41 \( 1 - 3.19iT - 41T^{2} \)
43 \( 1 + (-6.70 - 6.70i)T + 43iT^{2} \)
47 \( 1 + (7.34 - 7.34i)T - 47iT^{2} \)
53 \( 1 + (5.73 - 5.73i)T - 53iT^{2} \)
59 \( 1 + 8.68iT - 59T^{2} \)
61 \( 1 + 12.5iT - 61T^{2} \)
67 \( 1 + (1.87 - 1.87i)T - 67iT^{2} \)
71 \( 1 + 4.18iT - 71T^{2} \)
73 \( 1 + (-3.97 + 3.97i)T - 73iT^{2} \)
79 \( 1 + 9.66iT - 79T^{2} \)
83 \( 1 + (-0.585 + 0.585i)T - 83iT^{2} \)
89 \( 1 + 0.557T + 89T^{2} \)
97 \( 1 + (10.5 + 10.5i)T + 97iT^{2} \)
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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

−11.64407972798087263335288752535, −10.83860664162682893788697822797, −9.537608779115760383303076360560, −8.380436881673100926788727860400, −7.68604187997170889466556958696, −6.23523245302148857826013573669, −5.08087560288095423956996062709, −4.65277547994584241094379421986, −3.10494596779561324815423156999, −1.65369428608771081521267895679, 2.01192538055345542726572011914, 3.76561244299867539166445240742, 4.26862290249492515631875374053, 5.58435790798571019763557005890, 6.93975281967422160582864263873, 7.45844789831406383902381839870, 8.277697802642588878166592213864, 10.07565445109469093101507628982, 10.90204692074357578130686159945, 11.62271873493519726072929768076

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