Properties

Label 2-360-72.11-c1-0-42
Degree $2$
Conductor $360$
Sign $-0.447 + 0.894i$
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  + (−0.645 + 1.25i)2-s + (0.801 − 1.53i)3-s + (−1.16 − 1.62i)4-s + (0.5 + 0.866i)5-s + (1.41 + 1.99i)6-s + (−4.07 − 2.35i)7-s + (2.79 − 0.421i)8-s + (−1.71 − 2.46i)9-s + (−1.41 + 0.0704i)10-s + (−3.19 − 1.84i)11-s + (−3.42 + 0.489i)12-s + (−2.94 + 1.69i)13-s + (5.58 − 3.60i)14-s + (1.73 − 0.0730i)15-s + (−1.27 + 3.79i)16-s + 3.19i·17-s + ⋯
L(s)  = 1  + (−0.456 + 0.889i)2-s + (0.463 − 0.886i)3-s + (−0.583 − 0.812i)4-s + (0.223 + 0.387i)5-s + (0.577 + 0.816i)6-s + (−1.53 − 0.888i)7-s + (0.988 − 0.148i)8-s + (−0.571 − 0.820i)9-s + (−0.446 + 0.0222i)10-s + (−0.963 − 0.556i)11-s + (−0.989 + 0.141i)12-s + (−0.816 + 0.471i)13-s + (1.49 − 0.963i)14-s + (0.446 − 0.0188i)15-s + (−0.318 + 0.947i)16-s + 0.776i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.265424 - 0.429773i\)
\(L(\frac12)\) \(\approx\) \(0.265424 - 0.429773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.645 - 1.25i)T \)
3 \( 1 + (-0.801 + 1.53i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (4.07 + 2.35i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.19 + 1.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.94 - 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.19iT - 17T^{2} \)
19 \( 1 + 0.184T + 19T^{2} \)
23 \( 1 + (1.68 + 2.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.71 + 8.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.31 - 0.761i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.21iT - 37T^{2} \)
41 \( 1 + (-7.97 + 4.60i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.05 + 8.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.20 - 5.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.652T + 53T^{2} \)
59 \( 1 + (-7.25 + 4.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.59 + 3.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.725 + 1.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + (2.19 + 1.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.86 + 2.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.32iT - 89T^{2} \)
97 \( 1 + (8.59 - 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76964548004477993784036235341, −10.02145740610888883623074027652, −9.222251085642095779396455726321, −8.085240950585630579710369696063, −7.30711319139641984832473766315, −6.52406489644626530101818229300, −5.84593702020888112620954578968, −3.99392951770263801561059988643, −2.52269035780856262564041701934, −0.35328194288841821356847856185, 2.54608356181009410977636827263, 3.11851502706802184366353985284, 4.63500171402487716411270931482, 5.56547656832249138250212365736, 7.33056623501287624938423557723, 8.438066045374233287556680581612, 9.366596045285562383180197934791, 9.781562996514352130348912387442, 10.45788192687712937087607699589, 11.70310252115418032984288261487

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