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Properties

Label 2-360-120.59-c1-0-23
Degree $2$
Conductor $360$
Sign $-0.577 - 0.816i$
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  − 1.41i·2-s − 2.00·4-s − 2.23i·5-s − 5.16·7-s + 2.82i·8-s − 3.16·10-s + 3.05i·11-s + 0.837·13-s + 7.30i·14-s + 4.00·16-s − 6.32·19-s + 4.47i·20-s + 4.32·22-s − 4.47i·23-s − 5.00·25-s − 1.18i·26-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 0.999i·5-s − 1.95·7-s + 1.00i·8-s − 1.00·10-s + 0.921i·11-s + 0.232·13-s + 1.95i·14-s + 1.00·16-s − 1.45·19-s + 1.00i·20-s + 0.921·22-s − 0.932i·23-s − 1.00·25-s − 0.232i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.577 - 0.816i$
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: \((2,\ 360,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124369 + 0.240263i\)
\(L(\frac12)\) \(\approx\) \(0.124369 + 0.240263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
5 \( 1 + 2.23iT \)
good7 \( 1 + 5.16T + 7T^{2} \)
11 \( 1 - 3.05iT - 11T^{2} \)
13 \( 1 - 0.837T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + 5.42iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 18.8iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.61314830677779869096001373510, −10.00864364167786999827257864184, −9.130543438665355299696816084476, −8.542224948287722447535287943551, −6.95314543650627582541603963578, −5.79932049143313514064481845277, −4.51346266939693088869012836216, −3.58618667309477370457527973925, −2.15386693707151026946802445022, −0.17623747988585091616316784497, 3.08737558960831532393239133987, 3.86270586516254320614313577071, 5.70750720380320135461163522127, 6.43774406431362764392197294121, 6.96579269380390658844727598228, 8.233526803367150988863226753917, 9.244525874051015805639564303608, 10.03673126885952645597457266472, 10.85904512364903152659096106274, 12.25905851079599651310978717613

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