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Properties

Label 2-180-180.119-c1-0-0
Degree $2$
Conductor $180$
Sign $-0.205 - 0.978i$
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  + (−0.0443 − 1.41i)2-s + (−1.66 − 0.461i)3-s + (−1.99 + 0.125i)4-s + (−2.11 + 0.734i)5-s + (−0.578 + 2.38i)6-s + (0.667 − 1.15i)7-s + (0.265 + 2.81i)8-s + (2.57 + 1.54i)9-s + (1.13 + 2.95i)10-s + (−2.18 + 3.78i)11-s + (3.39 + 0.711i)12-s + (−3.56 + 2.05i)13-s + (−1.66 − 0.892i)14-s + (3.86 − 0.252i)15-s + (3.96 − 0.500i)16-s − 6.45·17-s + ⋯
L(s)  = 1  + (−0.0313 − 0.999i)2-s + (−0.963 − 0.266i)3-s + (−0.998 + 0.0626i)4-s + (−0.944 + 0.328i)5-s + (−0.236 + 0.971i)6-s + (0.252 − 0.436i)7-s + (0.0939 + 0.995i)8-s + (0.858 + 0.513i)9-s + (0.358 + 0.933i)10-s + (−0.659 + 1.14i)11-s + (0.978 + 0.205i)12-s + (−0.987 + 0.570i)13-s + (−0.444 − 0.238i)14-s + (0.997 − 0.0651i)15-s + (0.992 − 0.125i)16-s − 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.205 - 0.978i$
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: \((2,\ 180,\ (\ :1/2),\ -0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0346498 + 0.0426976i\)
\(L(\frac12)\) \(\approx\) \(0.0346498 + 0.0426976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0443 + 1.41i)T \)
3 \( 1 + (1.66 + 0.461i)T \)
5 \( 1 + (2.11 - 0.734i)T \)
good7 \( 1 + (-0.667 + 1.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.56 - 2.05i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.45T + 17T^{2} \)
19 \( 1 + 5.84iT - 19T^{2} \)
23 \( 1 + (-0.0875 + 0.0505i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.53 + 2.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.24iT - 37T^{2} \)
41 \( 1 + (-3.50 + 2.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.92 - 3.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.00 + 1.73i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.77T + 53T^{2} \)
59 \( 1 + (-1.37 - 2.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.216 + 0.374i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 + 7.28iT - 73T^{2} \)
79 \( 1 + (2.58 + 1.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.6 - 6.70i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.97iT - 89T^{2} \)
97 \( 1 + (2.08 + 1.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.69690727787367939501468991949, −11.76606585837248302076539775918, −11.12678013084396926645159088056, −10.38279466907877585921458775329, −9.203400432837426726730799284782, −7.64710538252253795534658260454, −6.92122106154157476918536112198, −4.87838596590026381417743065153, −4.33901914675347659085105222331, −2.27525139489465981595606831336, 0.05513938663084831715349839945, 3.86001459379990916867508444987, 5.06463163338252701401804190149, 5.82737174800176099058770190857, 7.18650566512226073256675453126, 8.135775213428610269126438624860, 9.132820249872200381992905104882, 10.46307453378364119808849848710, 11.40562627390839491085390595554, 12.48829747647636173566765249700

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