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Properties

Label 2-180-20.3-c1-0-11
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

\(L(1)\) \(\approx\) \(1.03947 - 1.13742i\)
\(L(\frac12)\) \(\approx\) \(1.03947 - 1.13742i\)
\(L(\frac{3}{2})\) 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

−12.28027943639995197037816891852, −11.64321854713640064014423908993, −10.72462583286486581610546272138, −9.376833417237356158778943578610, −8.663247039267933374849409520880, −7.01642705011643884916732879043, −5.67686025332234930635768142191, −4.55360371376143708591131710061, −3.48825090447479168940569400022, −1.43392854484617219200441845323, 3.03351974073920739973609272382, 4.00594085505945662383360458095, 5.68521253712848275746432641839, 6.46953689600195161302849812082, 7.80049789354900015256841351104, 8.337056271821739998998959790353, 10.10198157549735272147512258998, 11.12592721398720404943772405219, 12.04316356748259253012067595730, 13.16047999006397663681282872416

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