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Properties

Label 2-180-60.59-c1-0-9
Degree $2$
Conductor $180$
Sign $0.957 + 0.289i$
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  + 1.41·2-s + 2.00·4-s + (−0.707 − 2.12i)5-s + 2.82·8-s + (−1.00 − 3i)10-s + 6i·13-s + 4.00·16-s − 7.07·17-s + (−1.41 − 4.24i)20-s + (−3.99 + 3i)25-s + 8.48i·26-s − 4.24i·29-s + 5.65·32-s − 10.0·34-s − 12i·37-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + (−0.316 − 0.948i)5-s + 1.00·8-s + (−0.316 − 0.948i)10-s + 1.66i·13-s + 1.00·16-s − 1.71·17-s + (−0.316 − 0.948i)20-s + (−0.799 + 0.600i)25-s + 1.66i·26-s − 0.787i·29-s + 1.00·32-s − 1.71·34-s − 1.97i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91823 - 0.283763i\)
\(L(\frac12)\) \(\approx\) \(1.91823 - 0.283763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 2.12i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 4.24iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 4.24iT - 89T^{2} \)
97 \( 1 + 18iT - 97T^{2} \)
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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

−12.76764556788624936261815016013, −11.67098039836035768495031583259, −11.17817161410009431404671426026, −9.563863182414027096217474413831, −8.529260969187460784632796263764, −7.19730794675868632149030313265, −6.15840013348621533193404551450, −4.73416679443851427953696469932, −4.05305370462435972202077182232, −2.04458946909688645023042934820, 2.56008608997485602523465379538, 3.69268178623577564891669876375, 5.11329766835867295657521775591, 6.36662993213246602994347229996, 7.24031602159848404724320040389, 8.377999266228609552005626490576, 10.21938574872860454944465154213, 10.86469521173113201513945566499, 11.75629209410760694974832407251, 12.86071104125162242640732639284

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