Properties

Label 2-180-180.119-c1-0-20
Degree $2$
Conductor $180$
Sign $0.999 + 0.0134i$
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  + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (0.792 − 2.09i)5-s + 2.44i·6-s + (−1.5 + 2.59i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (1.99 + 2.44i)10-s + (2.12 − 3.67i)11-s + (−2.99 − 1.73i)12-s + (2.12 − 1.22i)13-s + (−2.12 − 3.67i)14-s + (−0.621 − 3.82i)15-s + (−2.00 + 3.46i)16-s + 1.41·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (0.866 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (0.354 − 0.935i)5-s + 0.999i·6-s + (−0.566 + 0.981i)7-s + 0.999·8-s + (0.5 − 0.866i)9-s + (0.632 + 0.774i)10-s + (0.639 − 1.10i)11-s + (−0.866 − 0.500i)12-s + (0.588 − 0.339i)13-s + (−0.566 − 0.981i)14-s + (−0.160 − 0.987i)15-s + (−0.500 + 0.866i)16-s + 0.342·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.999 + 0.0134i$
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: \((2,\ 180,\ (\ :1/2),\ 0.999 + 0.0134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18243 - 0.00792984i\)
\(L(\frac12)\) \(\approx\) \(1.18243 - 0.00792984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-0.792 + 2.09i)T \)
good7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 - 7.34iT - 19T^{2} \)
23 \( 1 + (4.5 - 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.36 - 3.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.44iT - 37T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.5 - 2.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 2.59i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.66iT - 89T^{2} \)
97 \( 1 + (-12.7 - 7.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.85193508997812214627072202289, −11.99022971525955601601015163872, −10.16385734055434550373558540963, −9.175443296019755223171705814247, −8.630964985319796978631497564217, −7.84004747951292344976554945082, −6.21892929127647967450222932393, −5.67776004011227768174042298481, −3.66977535277130217605698550385, −1.49637292456677036781826674845, 2.14378820246974677509595741166, 3.48054605144381372435472814821, 4.38427260930980732979795789538, 6.81676665586777729000675347994, 7.59856246335112219492338031685, 9.065888561669418752668682400240, 9.757577725312394277365377400484, 10.48284523038252874229390751607, 11.31896373644400899649378065772, 12.76047389329948893842270411304

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