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Properties

Label 2-180-36.23-c1-0-23
Degree $2$
Conductor $180$
Sign $-0.928 + 0.372i$
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  + (0.538 − 1.30i)2-s + (−0.398 − 1.68i)3-s + (−1.42 − 1.40i)4-s + (−0.866 − 0.5i)5-s + (−2.41 − 0.386i)6-s + (0.891 − 0.514i)7-s + (−2.60 + 1.09i)8-s + (−2.68 + 1.34i)9-s + (−1.12 + 0.863i)10-s + (1.84 + 3.20i)11-s + (−1.80 + 2.95i)12-s + (2.52 − 4.37i)13-s + (−0.193 − 1.44i)14-s + (−0.497 + 1.65i)15-s + (0.0339 + 3.99i)16-s − 2.05i·17-s + ⋯
L(s)  = 1  + (0.380 − 0.924i)2-s + (−0.230 − 0.973i)3-s + (−0.710 − 0.704i)4-s + (−0.387 − 0.223i)5-s + (−0.987 − 0.157i)6-s + (0.337 − 0.194i)7-s + (−0.921 + 0.388i)8-s + (−0.894 + 0.447i)9-s + (−0.354 + 0.272i)10-s + (0.557 + 0.965i)11-s + (−0.521 + 0.853i)12-s + (0.700 − 1.21i)13-s + (−0.0516 − 0.385i)14-s + (−0.128 + 0.428i)15-s + (0.00849 + 0.999i)16-s − 0.497i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.928 + 0.372i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.928 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212713 - 1.10177i\)
\(L(\frac12)\) \(\approx\) \(0.212713 - 1.10177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.538 + 1.30i)T \)
3 \( 1 + (0.398 + 1.68i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (-0.891 + 0.514i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.84 - 3.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.52 + 4.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.05iT - 17T^{2} \)
19 \( 1 + 5.65iT - 19T^{2} \)
23 \( 1 + (0.155 - 0.269i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.860 - 0.496i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.53 - 2.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.38T + 37T^{2} \)
41 \( 1 + (8.46 + 4.88i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.19 + 2.42i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.29 - 9.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.37iT - 53T^{2} \)
59 \( 1 + (-4.06 + 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.17 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.86 + 2.22i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.205T + 71T^{2} \)
73 \( 1 + 5.22T + 73T^{2} \)
79 \( 1 + (11.7 - 6.75i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.107 - 0.187i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18.2iT - 89T^{2} \)
97 \( 1 + (4.83 + 8.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.25637878491426906031497355470, −11.43371404119565459907575549768, −10.64055552929543534249501055550, −9.265664367638742405875390448422, −8.177406529037485269416864816035, −6.98544226426912941369549148575, −5.60897400731257630701378111675, −4.43601397241352051675209144016, −2.76186294797640997060632654364, −1.05719597929100698262724717539, 3.54717746393228781639282231255, 4.32492203880574715703114506540, 5.74810051116324093253028075058, 6.52351553480275235701182812950, 8.173516803459873408487332058256, 8.808415513068026873701635020344, 9.978935676438150914016658761731, 11.38391238034854021484130802431, 11.87195265607377488052709705129, 13.39786782260166641299635592446

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