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Properties

Label 2-180-180.119-c1-0-10
Degree $2$
Conductor $180$
Sign $0.543 - 0.839i$
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 + 0.0449i)2-s + (−1.62 + 0.589i)3-s + (1.99 + 0.126i)4-s + (0.0361 + 2.23i)5-s + (−2.32 + 0.759i)6-s + (−1.44 + 2.50i)7-s + (2.81 + 0.269i)8-s + (2.30 − 1.91i)9-s + (−0.0493 + 3.16i)10-s + (0.395 − 0.684i)11-s + (−3.32 + 0.968i)12-s + (4.04 − 2.33i)13-s + (−2.15 + 3.47i)14-s + (−1.37 − 3.62i)15-s + (3.96 + 0.506i)16-s − 5.89·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0317i)2-s + (−0.940 + 0.340i)3-s + (0.997 + 0.0634i)4-s + (0.0161 + 0.999i)5-s + (−0.950 + 0.310i)6-s + (−0.546 + 0.945i)7-s + (0.995 + 0.0951i)8-s + (0.768 − 0.639i)9-s + (−0.0156 + 0.999i)10-s + (0.119 − 0.206i)11-s + (−0.960 + 0.279i)12-s + (1.12 − 0.647i)13-s + (−0.575 + 0.927i)14-s + (−0.355 − 0.934i)15-s + (0.991 + 0.126i)16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.543 - 0.839i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.543 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38378 + 0.752171i\)
\(L(\frac12)\) \(\approx\) \(1.38378 + 0.752171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0449i)T \)
3 \( 1 + (1.62 - 0.589i)T \)
5 \( 1 + (-0.0361 - 2.23i)T \)
good7 \( 1 + (1.44 - 2.50i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.395 + 0.684i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.04 + 2.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 + 4.55iT - 19T^{2} \)
23 \( 1 + (-1.15 + 0.666i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.29 - 1.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.26 + 1.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.44iT - 37T^{2} \)
41 \( 1 + (5.91 - 3.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.95 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.60 + 1.50i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.70T + 53T^{2} \)
59 \( 1 + (-5.29 - 9.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.869 - 1.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.76 - 6.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.88T + 71T^{2} \)
73 \( 1 - 6.50iT - 73T^{2} \)
79 \( 1 + (3.30 + 1.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.33 + 4.81i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.00741iT - 89T^{2} \)
97 \( 1 + (5.74 + 3.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.84672362202917846957382899014, −11.72795137141935448466388976426, −11.07262887584858935399877396581, −10.35141390184446555760133337436, −8.865277704832682260289105879858, −7.01119761418089257387826001940, −6.29886831889744553672385064900, −5.47976975305730342245491514950, −4.00283190276502753739912254943, −2.69417608515139717225796474715, 1.46804030804442549039551119840, 3.96423594448413942337203232391, 4.78520394036886198714173491316, 6.15621171803424952485427286157, 6.80752706388830090920846364286, 8.168789525885131764451084514802, 9.824726945579069403634586283242, 10.89843202974495867301826622537, 11.69669747397040103572864908077, 12.65289273370485175368609224358

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