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Properties

Label 2-360-72.11-c1-0-3
Degree $2$
Conductor $360$
Sign $-0.991 - 0.132i$
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  + (−1.21 + 0.725i)2-s + (−0.452 + 1.67i)3-s + (0.946 − 1.76i)4-s + (0.5 + 0.866i)5-s + (−0.664 − 2.35i)6-s + (−0.550 − 0.317i)7-s + (0.131 + 2.82i)8-s + (−2.59 − 1.51i)9-s + (−1.23 − 0.688i)10-s + (4.49 + 2.59i)11-s + (2.51 + 2.37i)12-s + (−5.73 + 3.31i)13-s + (0.899 − 0.0138i)14-s + (−1.67 + 0.444i)15-s + (−2.21 − 3.33i)16-s + 4.95i·17-s + ⋯
L(s)  = 1  + (−0.858 + 0.513i)2-s + (−0.261 + 0.965i)3-s + (0.473 − 0.881i)4-s + (0.223 + 0.387i)5-s + (−0.271 − 0.962i)6-s + (−0.208 − 0.120i)7-s + (0.0463 + 0.998i)8-s + (−0.863 − 0.504i)9-s + (−0.390 − 0.217i)10-s + (1.35 + 0.782i)11-s + (0.727 + 0.686i)12-s + (−1.59 + 0.918i)13-s + (0.240 − 0.00371i)14-s + (−0.432 + 0.114i)15-s + (−0.552 − 0.833i)16-s + 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.991 - 0.132i$
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: \((2,\ 360,\ (\ :1/2),\ -0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0403795 + 0.607114i\)
\(L(\frac12)\) \(\approx\) \(0.0403795 + 0.607114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 - 0.725i)T \)
3 \( 1 + (0.452 - 1.67i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (0.550 + 0.317i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.49 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.73 - 3.31i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.95iT - 17T^{2} \)
19 \( 1 + 0.264T + 19T^{2} \)
23 \( 1 + (3.14 + 5.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.68 - 2.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.81 - 1.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.07iT - 37T^{2} \)
41 \( 1 + (9.24 - 5.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.19 + 3.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.26 - 5.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.24T + 53T^{2} \)
59 \( 1 + (0.776 - 0.448i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.09 - 3.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.19 - 9.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + (2.27 + 1.31i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.82 - 5.67i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.84iT - 89T^{2} \)
97 \( 1 + (-2.80 + 4.86i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.63816735522451196461653872715, −10.61535402451130676175205674581, −9.853051602510382597762244687387, −9.381014051372361914840554732078, −8.364755200942419551295654465082, −6.91893959890715372356663410425, −6.43469601935599963666103786524, −5.09032359014669256337593212259, −3.99781369999399357547052002152, −2.09887409324397531683029752086, 0.54606695686225490298622297577, 2.05731157855921055961822447065, 3.35384411491782104073129263973, 5.23967902661350581289149820235, 6.45173619372006522494260820525, 7.37694787978873901080689173048, 8.183814711030864859852781570351, 9.259094240881054396795494657838, 9.885957908995646587358798751703, 11.24649215989847604585194102101

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