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Properties

Label 2-180-20.3-c1-0-2
Degree $2$
Conductor $180$
Sign $0.0898 - 0.995i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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 + i)2-s − 2i·4-s + (1 + 2i)5-s + (2 + 2i)8-s + (−3 − i)10-s + (5 + 5i)13-s − 4·16-s + (−3 + 3i)17-s + (4 − 2i)20-s + (−3 + 4i)25-s − 10·26-s − 10i·29-s + (4 − 4i)32-s − 6i·34-s + (5 − 5i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − i·4-s + (0.447 + 0.894i)5-s + (0.707 + 0.707i)8-s + (−0.948 − 0.316i)10-s + (1.38 + 1.38i)13-s − 16-s + (−0.727 + 0.727i)17-s + (0.894 − 0.447i)20-s + (−0.600 + 0.800i)25-s − 1.96·26-s − 1.85i·29-s + (0.707 − 0.707i)32-s − 1.02i·34-s + (0.821 − 0.821i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.0898 - 0.995i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.0898 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656692 + 0.600142i\)
\(L(\frac12)\) \(\approx\) \(0.656692 + 0.600142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 \)
5 \( 1 + (-1 - 2i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-5 - 5i)T + 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 10iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (9 + 9i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (5 - 5i)T - 97iT^{2} \)
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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

−13.28805447705671232113784380851, −11.41149446976614744378539998100, −10.87792335298834723839209377984, −9.733438715358766573245468454021, −8.897159471699952949424650508928, −7.74049675756344181815633037748, −6.50151887304805599112061858185, −6.03101414069381520735005196939, −4.16612828926509674736220026343, −1.99879447334211626090229653412, 1.17790720201457529834445921806, 3.02669365898786424020353217412, 4.59652424833388629351632525704, 6.05281541199805411485597576184, 7.64530230168879810686107964962, 8.650779552603517603914584405882, 9.317463580516028159955219232479, 10.50438066563289087637408952687, 11.23982122252934138297467481386, 12.53586617313810056259335698502

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