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Properties

Label 2-180-60.47-c0-0-1
Degree $2$
Conductor $180$
Sign $0.662 + 0.749i$
Analytic cond. $0.0898317$
Root an. cond. $0.299719$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + 1.00i·10-s + (−1 + i)13-s − 1.00·16-s + (0.707 + 0.707i)20-s − 1.00i·25-s + 1.41i·26-s + 1.41·29-s + (−0.707 + 0.707i)32-s + (−1 − i)37-s + 1.00·40-s − 1.41i·41-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + 1.00i·10-s + (−1 + i)13-s − 1.00·16-s + (0.707 + 0.707i)20-s − 1.00i·25-s + 1.41i·26-s + 1.41·29-s + (−0.707 + 0.707i)32-s + (−1 − i)37-s + 1.00·40-s − 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(0.0898317\)
Root analytic conductor: \(0.299719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :0),\ 0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8274398088\)
\(L(\frac12)\) \(\approx\) \(0.8274398088\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−12.44189782341294048473951224601, −11.93250018415293239375197378971, −10.94870675399876954942065885747, −10.12464353733550797812134386447, −8.961480299107341667896798827893, −7.37277902640396022440918763588, −6.42809373412478744110268449053, −4.89767024677194076794894509282, −3.79394951927279425916884978760, −2.42384697399779368363070105446, 3.09402917711668797726896871776, 4.52548958960700959022514853505, 5.36330249790401760675189644478, 6.81023583686771266981051501347, 7.88776530148163649023129236316, 8.579235042579777653555315927582, 9.984500135610600031199816097305, 11.48231411481689417315861761561, 12.30989062683639107888882412815, 12.94900114826674432664013632073

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