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Properties

Label 2-180-45.13-c2-0-8
Degree $2$
Conductor $180$
Sign $0.417 + 0.908i$
Analytic cond. $4.90464$
Root an. cond. $2.21464$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.165 − 2.99i)3-s + (4.61 − 1.93i)5-s + (9.89 + 2.65i)7-s + (−8.94 − 0.991i)9-s + (−0.211 − 0.366i)11-s + (−8.51 + 2.28i)13-s + (−5.02 − 14.1i)15-s + (20.4 − 20.4i)17-s − 7.74i·19-s + (9.57 − 29.2i)21-s + (−28.8 + 7.71i)23-s + (17.5 − 17.8i)25-s + (−4.44 + 26.6i)27-s + (11.0 − 6.39i)29-s + (−11.5 + 19.9i)31-s + ⋯
L(s)  = 1  + (0.0551 − 0.998i)3-s + (0.922 − 0.386i)5-s + (1.41 + 0.378i)7-s + (−0.993 − 0.110i)9-s + (−0.0192 − 0.0333i)11-s + (−0.655 + 0.175i)13-s + (−0.334 − 0.942i)15-s + (1.20 − 1.20i)17-s − 0.407i·19-s + (0.456 − 1.39i)21-s + (−1.25 + 0.335i)23-s + (0.701 − 0.712i)25-s + (−0.164 + 0.986i)27-s + (0.382 − 0.220i)29-s + (−0.372 + 0.644i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.417 + 0.908i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ 0.417 + 0.908i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.58077 - 1.01329i\)
\(L(\frac12)\) \(\approx\) \(1.58077 - 1.01329i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.165 + 2.99i)T \)
5 \( 1 + (-4.61 + 1.93i)T \)
good7 \( 1 + (-9.89 - 2.65i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (0.211 + 0.366i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (8.51 - 2.28i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (-20.4 + 20.4i)T - 289iT^{2} \)
19 \( 1 + 7.74iT - 361T^{2} \)
23 \( 1 + (28.8 - 7.71i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (-11.0 + 6.39i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (11.5 - 19.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (40.0 - 40.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-28.9 + 50.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (17.5 - 65.5i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-4.49 - 1.20i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-15.2 - 15.2i)T + 2.80e3iT^{2} \)
59 \( 1 + (-30.0 - 17.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-38.1 - 66.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-25.6 - 95.6i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 21.5T + 5.04e3T^{2} \)
73 \( 1 + (82.7 + 82.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (38.0 - 21.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (5.14 - 19.2i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 17.7iT - 7.92e3T^{2} \)
97 \( 1 + (-120. - 32.2i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−12.07387605164675094715116239857, −11.70142687585148929684735516826, −10.22514005312337791088341109694, −9.050259613907068852681825637857, −8.107716449577653107738481491851, −7.15628278975350231614558264834, −5.73271746026120224649256500675, −4.95877156307672825669321252943, −2.54189630924980029518575541160, −1.35154531384916681853429708596, 2.00088231184310018698202343133, 3.76736824031154236515582317430, 5.06526469257275385196165270229, 5.93747688883220406623041646163, 7.66105095099360228330995532153, 8.593853686877399290105561702590, 9.973961062661223313618894854066, 10.38434693000535415079341610714, 11.36752864116785892514114119243, 12.51978315414619178020295396917

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