Properties

Label 2-360-15.8-c1-0-3
Degree $2$
Conductor $360$
Sign $0.920 + 0.391i$
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  + (2.12 − 0.707i)5-s − 5.65i·11-s + (3 + 3i)13-s + 4i·19-s + (2.82 − 2.82i)23-s + (3.99 − 3i)25-s + 1.41·29-s − 8·31-s + (7 − 7i)37-s + 1.41i·41-s + (4 + 4i)43-s + (2.82 + 2.82i)47-s + 7i·49-s + (−8.48 + 8.48i)53-s + (−4.00 − 12i)55-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)5-s − 1.70i·11-s + (0.832 + 0.832i)13-s + 0.917i·19-s + (0.589 − 0.589i)23-s + (0.799 − 0.600i)25-s + 0.262·29-s − 1.43·31-s + (1.15 − 1.15i)37-s + 0.220i·41-s + (0.609 + 0.609i)43-s + (0.412 + 0.412i)47-s + i·49-s + (−1.16 + 1.16i)53-s + (−0.539 − 1.61i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.920 + 0.391i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55524 - 0.316674i\)
\(L(\frac12)\) \(\approx\) \(1.55524 - 0.316674i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-7 + 7i)T - 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (-4 - 4i)T + 43iT^{2} \)
47 \( 1 + (-2.82 - 2.82i)T + 47iT^{2} \)
53 \( 1 + (8.48 - 8.48i)T - 53iT^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + (8 - 8i)T - 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + (11.3 - 11.3i)T - 83iT^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + (-5 + 5i)T - 97iT^{2} \)
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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

−11.11258693213806661799775732742, −10.69005442834392052734988030058, −9.264213847063539706177133930265, −8.879723870638715026049034426497, −7.71657750109353381973083640549, −6.15659129300509817029603398526, −5.87969600314991506112827772909, −4.36745726296670984490516728245, −3.00404297095822869316968961654, −1.35861733944311496550605413841, 1.73011782665181828286077374697, 3.07147214086244520619584582023, 4.65367612681756547262594712751, 5.65897159369838674556641163117, 6.75428581824896180174500678132, 7.57529067912904237890333773014, 8.939325073490663979612198584837, 9.699149049274288843695797933824, 10.50353258278012902588539281772, 11.34828666863556780290750736667

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