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Properties

Label 2-180-36.23-c1-0-12
Degree $2$
Conductor $180$
Sign $0.624 - 0.781i$
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  + (1.40 + 0.187i)2-s + (0.398 + 1.68i)3-s + (1.92 + 0.525i)4-s + (−0.866 − 0.5i)5-s + (0.242 + 2.43i)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 + (−0.117 + 3.46i)12-s + (2.52 − 4.37i)13-s + (−1.34 + 0.554i)14-s + (0.497 − 1.65i)15-s + (3.44 + 2.02i)16-s − 2.05i·17-s + ⋯
L(s)  = 1  + (0.991 + 0.132i)2-s + (0.230 + 0.973i)3-s + (0.964 + 0.262i)4-s + (−0.387 − 0.223i)5-s + (0.0990 + 0.995i)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 + (−0.0338 + 0.999i)12-s + (0.700 − 1.21i)13-s + (−0.359 + 0.148i)14-s + (0.128 − 0.428i)15-s + (0.861 + 0.507i)16-s − 0.497i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.624 - 0.781i$
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: \((2,\ 180,\ (\ :1/2),\ 0.624 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77668 + 0.854448i\)
\(L(\frac12)\) \(\approx\) \(1.77668 + 0.854448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.187i)T \)
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

−13.00080078182306119470429139725, −11.82169408684537448928584861406, −10.92008354183334433140700307170, −10.10527299485192772934053793647, −8.565265331548674063519082403160, −7.78765062539062473795383622632, −5.98106583962876524887774042708, −5.26996756516422254378482232032, −3.82655261185846160694298773260, −2.99364919681871623882096637969, 1.96952048776536602571778264253, 3.40465208921572855667654797046, 4.81022038266505742307744737301, 6.43287465891356943376078468771, 6.98092819997861447360381070867, 8.099611960915151806298731585778, 9.592722203672632274539365967898, 11.05483086273622644088395599656, 11.67575920885265222531287038288, 12.82199419344332260670019483125

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