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Properties

Label 2-180-20.7-c1-0-12
Degree $2$
Conductor $180$
Sign $0.307 + 0.951i$
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  + (1.19 − 0.760i)2-s + (0.844 − 1.81i)4-s + (−0.432 − 2.19i)5-s + (−0.611 + 0.611i)7-s + (−0.371 − 2.80i)8-s + (−2.18 − 2.28i)10-s + 5.12i·11-s + (1.76 − 1.76i)13-s + (−0.264 + 1.19i)14-s + (−2.57 − 3.06i)16-s + (3.76 + 3.76i)17-s + 1.22·19-s + (−4.34 − 1.06i)20-s + (3.89 + 6.11i)22-s + (−1.07 − 1.07i)23-s + ⋯
L(s)  = 1  + (0.843 − 0.537i)2-s + (0.422 − 0.906i)4-s + (−0.193 − 0.981i)5-s + (−0.231 + 0.231i)7-s + (−0.131 − 0.991i)8-s + (−0.690 − 0.723i)10-s + 1.54i·11-s + (0.488 − 0.488i)13-s + (−0.0706 + 0.319i)14-s + (−0.643 − 0.765i)16-s + (0.912 + 0.912i)17-s + 0.280·19-s + (−0.971 − 0.238i)20-s + (0.831 + 1.30i)22-s + (−0.224 − 0.224i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.307 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40002 - 1.01907i\)
\(L(\frac12)\) \(\approx\) \(1.40002 - 1.01907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.760i)T \)
3 \( 1 \)
5 \( 1 + (0.432 + 2.19i)T \)
good7 \( 1 + (0.611 - 0.611i)T - 7iT^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
13 \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \)
17 \( 1 + (-3.76 - 3.76i)T + 17iT^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + (1.07 + 1.07i)T + 23iT^{2} \)
29 \( 1 - 0.864iT - 29T^{2} \)
31 \( 1 - 7.81iT - 31T^{2} \)
37 \( 1 + (1.76 + 1.76i)T + 37iT^{2} \)
41 \( 1 + 5.52T + 41T^{2} \)
43 \( 1 + (6.20 + 6.20i)T + 43iT^{2} \)
47 \( 1 + (-2.29 + 2.29i)T - 47iT^{2} \)
53 \( 1 + (-2.62 + 2.62i)T - 53iT^{2} \)
59 \( 1 + 0.528T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (-6.20 + 6.20i)T - 67iT^{2} \)
71 \( 1 + 8.10iT - 71T^{2} \)
73 \( 1 + (2.25 - 2.25i)T - 73iT^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + (7.95 + 7.95i)T + 83iT^{2} \)
89 \( 1 - 7.25iT - 89T^{2} \)
97 \( 1 + (-0.793 - 0.793i)T + 97iT^{2} \)
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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

−12.44765162072276516111970844006, −11.94667747748964375249722095248, −10.48785249998122311784921948008, −9.742311926038661403660322767998, −8.500574689359979173892304161880, −7.10494824111785494056573601109, −5.70191104658883297541916544627, −4.76715708416320154624686582970, −3.54899748688465213528243580191, −1.65964952937428096901794708844, 2.95098694195355075172138316740, 3.84921557091148398694870737657, 5.56623656330314193974881475764, 6.46793810394955261973459951102, 7.47704613072921264889188853072, 8.486326396738225966645734319900, 10.00112740681417922401953134420, 11.38617838431468609852399851364, 11.63028930271591429425122950233, 13.22727555810358714501642760347

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