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Properties

Label 2-360-40.27-c1-0-7
Degree $2$
Conductor $360$
Sign $0.973 - 0.229i$
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  − 1.41·2-s + 2.00·4-s + (−2.12 − 0.707i)5-s + (−1 + i)7-s − 2.82·8-s + (3 + 1.00i)10-s + 4.24·11-s + (2 + 2i)13-s + (1.41 − 1.41i)14-s + 4.00·16-s + (2.82 + 2.82i)17-s − 6i·19-s + (−4.24 − 1.41i)20-s − 6·22-s + (1.41 + 1.41i)23-s + ⋯
L(s)  = 1  − 1.00·2-s + 1.00·4-s + (−0.948 − 0.316i)5-s + (−0.377 + 0.377i)7-s − 1.00·8-s + (0.948 + 0.316i)10-s + 1.27·11-s + (0.554 + 0.554i)13-s + (0.377 − 0.377i)14-s + 1.00·16-s + (0.685 + 0.685i)17-s − 1.37i·19-s + (−0.948 − 0.316i)20-s − 1.27·22-s + (0.294 + 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771361 + 0.0898125i\)
\(L(\frac12)\) \(\approx\) \(0.771361 + 0.0898125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
5 \( 1 + (2.12 + 0.707i)T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (-1.41 - 1.41i)T + 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + (-8 + 8i)T - 37iT^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + (1.41 - 1.41i)T - 47iT^{2} \)
53 \( 1 + (-1.41 - 1.41i)T + 53iT^{2} \)
59 \( 1 - 9.89iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (8 + 8i)T + 67iT^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (5 - 5i)T - 73iT^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \)
89 \( 1 + 2.82iT - 89T^{2} \)
97 \( 1 + (-7 - 7i)T + 97iT^{2} \)
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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

−11.42571056525182551912813590628, −10.62782384516897712784600638085, −9.115353523988109771265672068658, −9.059406046238057135120127574861, −7.83342757766155652460779674975, −6.89539450830188497751019677430, −6.00363396358678831166323670719, −4.28798641067122387119230957168, −3.05286357285641959687234866477, −1.15889723076161050698592358299, 0.979125829850866179897914691181, 3.04720932935805741441554362562, 4.04222774060526623805799186903, 5.98389512477708345272884304268, 6.85319454372157449217379154168, 7.78663770876390656019263073713, 8.502767435173733656612246045575, 9.664730486923846662680560111356, 10.34107867829668658610913566855, 11.44498592491246723425512092989

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