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Properties

Label 2-360-40.3-c1-0-16
Degree $2$
Conductor $360$
Sign $-0.0530 + 0.998i$
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  + (−0.512 − 1.31i)2-s + (−1.47 + 1.35i)4-s + (1.83 + 1.28i)5-s + (−2.94 − 2.94i)7-s + (2.53 + 1.25i)8-s + (0.754 − 3.07i)10-s + 1.61·11-s + (2.50 − 2.50i)13-s + (−2.37 + 5.39i)14-s + (0.350 − 3.98i)16-s + (4.59 − 4.59i)17-s − 4i·19-s + (−4.43 + 0.578i)20-s + (−0.825 − 2.12i)22-s + (1.09 − 1.09i)23-s + ⋯
L(s)  = 1  + (−0.362 − 0.932i)2-s + (−0.737 + 0.675i)4-s + (0.818 + 0.574i)5-s + (−1.11 − 1.11i)7-s + (0.896 + 0.442i)8-s + (0.238 − 0.971i)10-s + 0.485·11-s + (0.696 − 0.696i)13-s + (−0.635 + 1.44i)14-s + (0.0876 − 0.996i)16-s + (1.11 − 1.11i)17-s − 0.917i·19-s + (−0.991 + 0.129i)20-s + (−0.176 − 0.452i)22-s + (0.227 − 0.227i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0530 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0530 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0530 + 0.998i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.0530 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761851 - 0.803424i\)
\(L(\frac12)\) \(\approx\) \(0.761851 - 0.803424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.512 + 1.31i)T \)
3 \( 1 \)
5 \( 1 + (-1.83 - 1.28i)T \)
good7 \( 1 + (2.94 + 2.94i)T + 7iT^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 + (-2.50 + 2.50i)T - 13iT^{2} \)
17 \( 1 + (-4.59 + 4.59i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-1.09 + 1.09i)T - 23iT^{2} \)
29 \( 1 - 4.75T + 29T^{2} \)
31 \( 1 - 5.01iT - 31T^{2} \)
37 \( 1 + (2.50 + 2.50i)T + 37iT^{2} \)
41 \( 1 + 9.18T + 41T^{2} \)
43 \( 1 + (7.40 + 7.40i)T + 43iT^{2} \)
47 \( 1 + (-7.32 - 7.32i)T + 47iT^{2} \)
53 \( 1 + (3.11 - 3.11i)T - 53iT^{2} \)
59 \( 1 - 1.61iT - 59T^{2} \)
61 \( 1 - 6.78iT - 61T^{2} \)
67 \( 1 + (-7.40 + 7.40i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (-7.57 - 7.57i)T + 83iT^{2} \)
89 \( 1 - 2.74iT - 89T^{2} \)
97 \( 1 + (-2.40 + 2.40i)T - 97iT^{2} \)
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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

−10.89681423011437542402223237244, −10.29249684811745507290892643212, −9.679035300941832313539618446918, −8.768779980881076225778366812873, −7.35026939482412930464086633499, −6.60792510481149574336554124686, −5.14866106221003501110501232697, −3.59143232294683204103218256703, −2.88908348318522249619257616733, −0.979278526804446615595442368149, 1.62080084818696585843394861104, 3.66259261004435778835970037341, 5.21541231067385948414745110715, 6.11436751111714099032432013441, 6.52342492850633067606442866039, 8.209575507929630786451488714531, 8.826098900807513689835538944650, 9.698602489973680956483128121428, 10.20336798419528360619751880924, 11.87950915967736234743656151424

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