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Properties

Label 2-360-40.29-c1-0-3
Degree $2$
Conductor $360$
Sign $-1$
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 − 2.82i·8-s − 3.16·10-s + 4.47i·11-s − 6.32·13-s + 4.00·16-s + 2.82i·17-s − 4.47i·20-s − 6.32·22-s − 5.65i·23-s − 5.00·25-s − 8.94i·26-s + 4.47i·29-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + 0.999i·5-s − 1.00i·8-s − 1.00·10-s + 1.34i·11-s − 1.75·13-s + 1.00·16-s + 0.685i·17-s − 1.00i·20-s − 1.34·22-s − 1.17i·23-s − 1.00·25-s − 1.75i·26-s + 0.830i·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871596i\)
\(L(\frac12)\) \(\approx\) \(0.871596i\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + 6.32T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 6.32T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 - 11.3iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 4.47iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
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

−12.28104293146943353882807192811, −10.63824593649506550032805556632, −10.00375414201745315431097190234, −9.132760901259212823741114879339, −7.77472659363087873887294107987, −7.17994434650798021870876130076, −6.40060107780927609488789468359, −5.10082289346785253358422820986, −4.13296508954357498512058083675, −2.48510880608957523542355183570, 0.59134674158970476226223242617, 2.34041013222452959040818539994, 3.71032303271125624701282049567, 4.93550850355317429632472861761, 5.63888434829617977217649930972, 7.48433071739903033841009052676, 8.456800261088512679776161053423, 9.321631318963208368701762072805, 9.968399033688444135131593502054, 11.16175302533116686186790739160

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