Properties

Label 2-180-180.7-c1-0-18
Degree $2$
Conductor $180$
Sign $0.444 + 0.895i$
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  + (−1.35 − 0.416i)2-s + (1.69 + 0.344i)3-s + (1.65 + 1.12i)4-s + (−1.18 − 1.89i)5-s + (−2.15 − 1.17i)6-s + (−1.14 − 4.28i)7-s + (−1.76 − 2.21i)8-s + (2.76 + 1.17i)9-s + (0.807 + 3.05i)10-s + (−0.725 − 0.418i)11-s + (2.41 + 2.48i)12-s + (5.13 + 1.37i)13-s + (−0.234 + 6.26i)14-s + (−1.35 − 3.62i)15-s + (1.46 + 3.72i)16-s + (−0.00689 − 0.00689i)17-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (0.979 + 0.199i)3-s + (0.826 + 0.563i)4-s + (−0.528 − 0.848i)5-s + (−0.877 − 0.479i)6-s + (−0.433 − 1.61i)7-s + (−0.623 − 0.781i)8-s + (0.920 + 0.390i)9-s + (0.255 + 0.966i)10-s + (−0.218 − 0.126i)11-s + (0.697 + 0.716i)12-s + (1.42 + 0.381i)13-s + (−0.0626 + 1.67i)14-s + (−0.349 − 0.936i)15-s + (0.365 + 0.930i)16-s + (−0.00167 − 0.00167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816445 - 0.506154i\)
\(L(\frac12)\) \(\approx\) \(0.816445 - 0.506154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.416i)T \)
3 \( 1 + (-1.69 - 0.344i)T \)
5 \( 1 + (1.18 + 1.89i)T \)
good7 \( 1 + (1.14 + 4.28i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.725 + 0.418i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.13 - 1.37i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.00689 + 0.00689i)T + 17iT^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + (-0.595 + 2.22i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.36 - 2.52i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.87 - 2.81i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.08 + 3.08i)T + 37iT^{2} \)
41 \( 1 + (-4.71 - 8.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.250 + 0.0670i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.147 - 0.549i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.01 - 6.01i)T - 53iT^{2} \)
59 \( 1 + (-1.15 - 1.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.01 + 5.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 0.830i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.16iT - 71T^{2} \)
73 \( 1 + (-1.15 + 1.15i)T - 73iT^{2} \)
79 \( 1 + (-4.71 + 8.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.73 - 0.732i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 0.849iT - 89T^{2} \)
97 \( 1 + (9.71 - 2.60i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.64528544872608718433864315721, −11.10471633432402138139704244835, −10.43464398286920037415925266666, −9.324657175815922195898923963013, −8.512689741077856455419268895781, −7.69948396349078548089071416437, −6.69520235728165534323464955545, −4.25944251088845643342996425389, −3.40031469100945993463154238708, −1.22227491835063504530639994288, 2.25951866347499892101252669402, 3.36879392063593225408976309821, 5.84318108836475949310787972091, 6.79741416686705345014374064025, 8.009770965759711421920784149455, 8.663134291225213140101471579694, 9.564595479263843904496481801483, 10.67398369329092960889397655607, 11.71495655977600361729355058482, 12.74134992178398699434189886041

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