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Properties

Label 2-180-9.4-c1-0-2
Degree $2$
Conductor $180$
Sign $-0.0707 + 0.997i$
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  + (−1.71 − 0.211i)3-s + (0.5 − 0.866i)5-s + (−1.36 − 2.36i)7-s + (2.91 + 0.728i)9-s + (−2.76 − 4.78i)11-s + (1.76 − 3.05i)13-s + (−1.04 + 1.38i)15-s − 5.52·17-s + 7.52·19-s + (1.84 + 4.36i)21-s + (−0.367 + 0.635i)23-s + (−0.499 − 0.866i)25-s + (−4.84 − 1.86i)27-s + (2.23 + 3.86i)29-s + (−3.76 + 6.51i)31-s + ⋯
L(s)  = 1  + (−0.992 − 0.122i)3-s + (0.223 − 0.387i)5-s + (−0.516 − 0.894i)7-s + (0.970 + 0.242i)9-s + (−0.832 − 1.44i)11-s + (0.488 − 0.846i)13-s + (−0.269 + 0.357i)15-s − 1.33·17-s + 1.72·19-s + (0.403 + 0.951i)21-s + (−0.0765 + 0.132i)23-s + (−0.0999 − 0.173i)25-s + (−0.933 − 0.359i)27-s + (0.414 + 0.718i)29-s + (−0.675 + 1.17i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0707 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0707 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.0707 + 0.997i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.0707 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504846 - 0.541924i\)
\(L(\frac12)\) \(\approx\) \(0.504846 - 0.541924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.71 + 0.211i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1.36 + 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.76 + 4.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.76 + 3.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 - 7.52T + 19T^{2} \)
23 \( 1 + (0.367 - 0.635i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.23 - 3.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.76 - 6.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.05T + 37T^{2} \)
41 \( 1 + (-0.527 + 0.914i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.76 - 3.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.604 + 1.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.46T + 53T^{2} \)
59 \( 1 + (-0.734 + 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.52 + 7.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.12 + 3.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.63 - 4.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (4.73 + 8.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.52531139044933317438226042198, −11.14505034612412716359711522719, −10.72112428700727658516303206763, −9.596120803718972415746126101278, −8.227146607129655226333488132327, −7.05711466525077275081759022314, −5.92989273194391076190689653579, −5.01706422192663827601561701501, −3.39277919791437035991384470212, −0.77685897517753579213527893273, 2.31065613372970975136192187267, 4.32932911038998799069014948341, 5.53106745391343065617823930504, 6.52651055178820817649129282599, 7.49829869175354370560416316899, 9.292038352778389827702638357428, 9.880924304261842529819496028368, 11.08497359627426027617704275854, 11.86299507376217033285156576209, 12.77263948601625979627686615036

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