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Properties

Label 2-180-180.103-c1-0-2
Degree $2$
Conductor $180$
Sign $-0.911 - 0.410i$
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  + (−1.16 + 0.801i)2-s + (−1.70 + 0.286i)3-s + (0.715 − 1.86i)4-s + (1.09 + 1.94i)5-s + (1.76 − 1.70i)6-s + (−0.0500 + 0.186i)7-s + (0.662 + 2.74i)8-s + (2.83 − 0.979i)9-s + (−2.84 − 1.39i)10-s + (−4.51 + 2.60i)11-s + (−0.687 + 3.39i)12-s + (−6.23 + 1.67i)13-s + (−0.0913 − 0.257i)14-s + (−2.43 − 3.01i)15-s + (−2.97 − 2.67i)16-s + (−0.305 + 0.305i)17-s + ⋯
L(s)  = 1  + (−0.823 + 0.566i)2-s + (−0.986 + 0.165i)3-s + (0.357 − 0.933i)4-s + (0.491 + 0.871i)5-s + (0.718 − 0.695i)6-s + (−0.0189 + 0.0706i)7-s + (0.234 + 0.972i)8-s + (0.945 − 0.326i)9-s + (−0.898 − 0.439i)10-s + (−1.36 + 0.785i)11-s + (−0.198 + 0.980i)12-s + (−1.72 + 0.463i)13-s + (−0.0244 − 0.0689i)14-s + (−0.628 − 0.777i)15-s + (−0.743 − 0.668i)16-s + (−0.0740 + 0.0740i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.911 - 0.410i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0850625 + 0.396408i\)
\(L(\frac12)\) \(\approx\) \(0.0850625 + 0.396408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.801i)T \)
3 \( 1 + (1.70 - 0.286i)T \)
5 \( 1 + (-1.09 - 1.94i)T \)
good7 \( 1 + (0.0500 - 0.186i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.51 - 2.60i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.23 - 1.67i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.305 - 0.305i)T - 17iT^{2} \)
19 \( 1 - 2.39T + 19T^{2} \)
23 \( 1 + (-1.13 - 4.24i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.28 - 1.31i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.124 - 0.0717i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.83 + 2.83i)T - 37iT^{2} \)
41 \( 1 + (1.76 - 3.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.96 - 2.13i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.07 + 7.73i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.86 + 3.86i)T + 53iT^{2} \)
59 \( 1 + (0.587 - 1.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.54 - 4.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.12 + 2.44i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.80iT - 71T^{2} \)
73 \( 1 + (1.96 + 1.96i)T + 73iT^{2} \)
79 \( 1 + (-1.89 - 3.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.57 + 2.03i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 13.0iT - 89T^{2} \)
97 \( 1 + (5.44 + 1.45i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.01264253996746467797277852617, −11.80059640280094798134224033690, −10.83997690454648993030443732091, −9.994360504063092146640369192226, −9.513330201231952186426306856321, −7.47785918918586381750826905547, −7.12652223651323707335058148135, −5.75521690044627165354768286167, −4.93749995041233608617507766930, −2.28405079202752012412731446617, 0.51072981937686292872288853537, 2.48544334518533961458149402244, 4.69894171819607409756063038257, 5.72417298567498088800473937136, 7.27646387184152414368463130795, 8.157737600172002121422643577627, 9.512069534834731320216801448117, 10.26455184476084460355972699575, 11.11281103505066708203921666346, 12.32473489818838904464364018256

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