Properties

Label 2-360-120.77-c1-0-9
Degree $2$
Conductor $360$
Sign $-0.0649 - 0.997i$
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  + (0.760 + 1.19i)2-s + (−0.844 + 1.81i)4-s + (2.23 + 0.0113i)5-s + (0.471 + 0.471i)7-s + (−2.80 + 0.372i)8-s + (1.68 + 2.67i)10-s − 0.335·11-s + (3.50 + 3.50i)13-s + (−0.203 + 0.921i)14-s + (−2.57 − 3.06i)16-s + (−2.53 + 2.53i)17-s + 4.07·19-s + (−1.90 + 4.04i)20-s + (−0.255 − 0.400i)22-s + (−6.20 − 6.20i)23-s + ⋯
L(s)  = 1  + (0.537 + 0.843i)2-s + (−0.422 + 0.906i)4-s + (0.999 + 0.00506i)5-s + (0.178 + 0.178i)7-s + (−0.991 + 0.131i)8-s + (0.533 + 0.845i)10-s − 0.101·11-s + (0.971 + 0.971i)13-s + (−0.0545 + 0.246i)14-s + (−0.643 − 0.765i)16-s + (−0.614 + 0.614i)17-s + 0.934·19-s + (−0.426 + 0.904i)20-s + (−0.0544 − 0.0853i)22-s + (−1.29 − 1.29i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0649 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0649 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0649 - 0.997i$
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: \((2,\ 360,\ (\ :1/2),\ -0.0649 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32354 + 1.41242i\)
\(L(\frac12)\) \(\approx\) \(1.32354 + 1.41242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.760 - 1.19i)T \)
3 \( 1 \)
5 \( 1 + (-2.23 - 0.0113i)T \)
good7 \( 1 + (-0.471 - 0.471i)T + 7iT^{2} \)
11 \( 1 + 0.335T + 11T^{2} \)
13 \( 1 + (-3.50 - 3.50i)T + 13iT^{2} \)
17 \( 1 + (2.53 - 2.53i)T - 17iT^{2} \)
19 \( 1 - 4.07T + 19T^{2} \)
23 \( 1 + (6.20 + 6.20i)T + 23iT^{2} \)
29 \( 1 + 2.42iT - 29T^{2} \)
31 \( 1 + 6.41T + 31T^{2} \)
37 \( 1 + (-2.24 + 2.24i)T - 37iT^{2} \)
41 \( 1 - 5.80iT - 41T^{2} \)
43 \( 1 + (-4.87 - 4.87i)T + 43iT^{2} \)
47 \( 1 + (-1.68 + 1.68i)T - 47iT^{2} \)
53 \( 1 + (-3.05 + 3.05i)T - 53iT^{2} \)
59 \( 1 + 12.2iT - 59T^{2} \)
61 \( 1 + 7.49iT - 61T^{2} \)
67 \( 1 + (5.55 - 5.55i)T - 67iT^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (-5.05 + 5.05i)T - 73iT^{2} \)
79 \( 1 + 8.85iT - 79T^{2} \)
83 \( 1 + (4.78 - 4.78i)T - 83iT^{2} \)
89 \( 1 - 8.33T + 89T^{2} \)
97 \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \)
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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

−11.84595356770186400572323812177, −10.82848067487092664596320176392, −9.568958485587040418413755029655, −8.823042469027458969667183520319, −7.88788417832351418194910256086, −6.54094173912290826373735381246, −6.06628948386611032892438935040, −4.92317217096120370678099830749, −3.78686189323418355439884141117, −2.15226449667547900774287369511, 1.34594068256164469963114550044, 2.73863980103707427796824815457, 3.96066880397805871781241350513, 5.41896463087336217688119975742, 5.85839076753456294073836365855, 7.32047045120561395262405741397, 8.778351374568840227114016312228, 9.575517533451997495698070694712, 10.44080769777249932723234044440, 11.12541284612692924809174349073

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