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Properties

Label 2-360-120.59-c1-0-3
Degree $2$
Conductor $360$
Sign $-0.985 - 0.169i$
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  + (0.927 + 1.06i)2-s + (−0.280 + 1.98i)4-s + (−2.18 + 0.468i)5-s − 3.02·7-s + (−2.37 + 1.53i)8-s + (−2.52 − 1.90i)10-s + 3.62i·11-s + 1.69·13-s + (−2.80 − 3.22i)14-s + (−3.84 − 1.11i)16-s − 6.60·17-s + 5.12·19-s + (−0.313 − 4.46i)20-s + (−3.86 + 3.35i)22-s + 6.67i·23-s + ⋯
L(s)  = 1  + (0.655 + 0.755i)2-s + (−0.140 + 0.990i)4-s + (−0.977 + 0.209i)5-s − 1.14·7-s + (−0.839 + 0.543i)8-s + (−0.799 − 0.601i)10-s + 1.09i·11-s + 0.470·13-s + (−0.748 − 0.862i)14-s + (−0.960 − 0.277i)16-s − 1.60·17-s + 1.17·19-s + (−0.0700 − 0.997i)20-s + (−0.824 + 0.716i)22-s + 1.39i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0853911 + 1.00072i\)
\(L(\frac12)\) \(\approx\) \(0.0853911 + 1.00072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.927 - 1.06i)T \)
3 \( 1 \)
5 \( 1 + (2.18 - 0.468i)T \)
good7 \( 1 + 3.02T + 7T^{2} \)
11 \( 1 - 3.62iT - 11T^{2} \)
13 \( 1 - 1.69T + 13T^{2} \)
17 \( 1 + 6.60T + 17T^{2} \)
19 \( 1 - 5.12T + 19T^{2} \)
23 \( 1 - 6.67iT - 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 0.371T + 37T^{2} \)
41 \( 1 - 5.83iT - 41T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 - 0.525iT - 47T^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 - 4.86iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + 2.45T + 71T^{2} \)
73 \( 1 + 14.5iT - 73T^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 - 5.79T + 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 - 9.33iT - 97T^{2} \)
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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

−11.95123385248277604810381061731, −11.32934770240250202356394046727, −9.884250526577154754359127636971, −8.961756199444896999410084728377, −7.83523755901233362957652779477, −7.00756308083384635707393211822, −6.33318686540806826871362783010, −4.89411837356652522219958835511, −3.91398066314068413538903690294, −2.90807940082637226483979586365, 0.54853001880196575795502505780, 2.83561634224986253914388939971, 3.69866118062682136003653068504, 4.75479284056672982503204191589, 6.10426730268104657162441786048, 6.89193276134901938742028675849, 8.514045595792838479555274513275, 9.160489863808419395876491007815, 10.44131988543961343324070672308, 11.08510357753680986108080781598

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