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Properties

Label 2-180-180.43-c1-0-20
Degree $2$
Conductor $180$
Sign $0.831 + 0.554i$
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  + (1.33 − 0.453i)2-s + (−1.51 + 0.835i)3-s + (1.58 − 1.21i)4-s + (1.01 − 1.99i)5-s + (−1.65 + 1.80i)6-s + (−0.00373 + 0.00100i)7-s + (1.57 − 2.34i)8-s + (1.60 − 2.53i)9-s + (0.458 − 3.12i)10-s + (3.58 + 2.07i)11-s + (−1.39 + 3.17i)12-s + (−0.767 + 2.86i)13-s + (−0.00455 + 0.00303i)14-s + (0.121 + 3.87i)15-s + (1.05 − 3.85i)16-s + (−2.07 + 2.07i)17-s + ⋯
L(s)  = 1  + (0.947 − 0.320i)2-s + (−0.876 + 0.482i)3-s + (0.794 − 0.607i)4-s + (0.454 − 0.890i)5-s + (−0.675 + 0.737i)6-s + (−0.00141 + 0.000378i)7-s + (0.558 − 0.829i)8-s + (0.534 − 0.844i)9-s + (0.145 − 0.989i)10-s + (1.08 + 0.624i)11-s + (−0.403 + 0.915i)12-s + (−0.212 + 0.794i)13-s + (−0.00121 + 0.000811i)14-s + (0.0313 + 0.999i)15-s + (0.262 − 0.964i)16-s + (−0.503 + 0.503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.831 + 0.554i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.831 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58892 - 0.481194i\)
\(L(\frac12)\) \(\approx\) \(1.58892 - 0.481194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.453i)T \)
3 \( 1 + (1.51 - 0.835i)T \)
5 \( 1 + (-1.01 + 1.99i)T \)
good7 \( 1 + (0.00373 - 0.00100i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.58 - 2.07i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.767 - 2.86i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.07 - 2.07i)T - 17iT^{2} \)
19 \( 1 + 7.31T + 19T^{2} \)
23 \( 1 + (3.73 + 1.00i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.99 - 2.88i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.77 + 1.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.09 - 6.09i)T - 37iT^{2} \)
41 \( 1 + (-1.82 - 3.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.259 + 0.968i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.55 + 1.22i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.92 - 6.92i)T + 53iT^{2} \)
59 \( 1 + (-2.76 - 4.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.59 + 6.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.851 - 3.17i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (6.50 + 6.50i)T + 73iT^{2} \)
79 \( 1 + (-2.99 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.756 + 2.82i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 9.18iT - 89T^{2} \)
97 \( 1 + (-0.607 - 2.26i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.29012193188613551059255153902, −11.97913883178597233237550055909, −10.71706334862284754997372666236, −9.873423885090028830238903642157, −8.816360585835212061193374144273, −6.71166471620681114932548088824, −6.09551091954383918462363326719, −4.66059578082090140270359333412, −4.18830335113509858819542348843, −1.73663936764223112936335076806, 2.31252158702569400449785584799, 4.01448234168515687428241047127, 5.51504275145157386111598871396, 6.38881149501819944225515698460, 7.01733770766449807076364573136, 8.359723381663184402363508162285, 10.23130785289761233045552880276, 11.05660622043790159647675621775, 11.85080492903278679126052663475, 12.80244080271873820820057440844

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