Properties

Label 2-360-40.3-c1-0-20
Degree $2$
Conductor $360$
Sign $0.865 + 0.501i$
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  + (1.17 − 0.788i)2-s + (0.758 − 1.85i)4-s + (−0.386 + 2.20i)5-s + (1.51 + 1.51i)7-s + (−0.568 − 2.77i)8-s + (1.28 + 2.89i)10-s + 3.92·11-s + (3.56 − 3.56i)13-s + (2.97 + 0.585i)14-s + (−2.85 − 2.80i)16-s + (−1.37 + 1.37i)17-s − 4i·19-s + (3.78 + 2.38i)20-s + (4.60 − 3.09i)22-s + (−5.17 + 5.17i)23-s + ⋯
L(s)  = 1  + (0.830 − 0.557i)2-s + (0.379 − 0.925i)4-s + (−0.172 + 0.984i)5-s + (0.573 + 0.573i)7-s + (−0.200 − 0.979i)8-s + (0.405 + 0.914i)10-s + 1.18·11-s + (0.988 − 0.988i)13-s + (0.795 + 0.156i)14-s + (−0.712 − 0.701i)16-s + (−0.333 + 0.333i)17-s − 0.917i·19-s + (0.846 + 0.533i)20-s + (0.982 − 0.659i)22-s + (−1.07 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.865 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15604 - 0.579211i\)
\(L(\frac12)\) \(\approx\) \(2.15604 - 0.579211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.788i)T \)
3 \( 1 \)
5 \( 1 + (0.386 - 2.20i)T \)
good7 \( 1 + (-1.51 - 1.51i)T + 7iT^{2} \)
11 \( 1 - 3.92T + 11T^{2} \)
13 \( 1 + (-3.56 + 3.56i)T - 13iT^{2} \)
17 \( 1 + (1.37 - 1.37i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (5.17 - 5.17i)T - 23iT^{2} \)
29 \( 1 + 5.95T + 29T^{2} \)
31 \( 1 - 7.12iT - 31T^{2} \)
37 \( 1 + (3.56 + 3.56i)T + 37iT^{2} \)
41 \( 1 - 2.75T + 41T^{2} \)
43 \( 1 + (-5.40 - 5.40i)T + 43iT^{2} \)
47 \( 1 + (1.54 + 1.54i)T + 47iT^{2} \)
53 \( 1 + (1.81 - 1.81i)T - 53iT^{2} \)
59 \( 1 - 3.92iT - 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (5.40 - 5.40i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 7.12T + 79T^{2} \)
83 \( 1 + (6.67 + 6.67i)T + 83iT^{2} \)
89 \( 1 + 18.4iT - 89T^{2} \)
97 \( 1 + (10.4 - 10.4i)T - 97iT^{2} \)
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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

−11.29024024587278309337189533984, −10.89637687240198607005605828636, −9.764766965555861392439342056860, −8.697113758084946467308186206686, −7.35893353073280717285010119125, −6.29916664799389268772735248662, −5.52971677656390901840339748716, −4.06633470121126682078310667973, −3.17864467275976402677148013726, −1.75249608383752038717588724173, 1.72098773052178247465044446417, 4.08192159048372531139262465458, 4.18284713205037151134031267811, 5.68366817251961730240019879157, 6.57489690001677347825723768443, 7.72616564304433268375289904846, 8.545040805429322670168180992564, 9.388338948486197499733229742916, 10.98693606632160998576872215538, 11.75687372308894300012410063117

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