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Properties

Label 2-180-45.22-c2-0-9
Degree $2$
Conductor $180$
Sign $0.915 + 0.402i$
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  + (2.46 + 1.70i)3-s + (2.19 − 4.49i)5-s + (2.81 − 10.5i)7-s + (3.17 + 8.42i)9-s + (−2.64 − 4.57i)11-s + (−2.60 − 9.70i)13-s + (13.0 − 7.32i)15-s + (6.35 + 6.35i)17-s + 28.5i·19-s + (24.8 − 21.1i)21-s + (10.4 + 38.9i)23-s + (−15.3 − 19.7i)25-s + (−6.54 + 26.1i)27-s + (−2.28 + 1.32i)29-s + (24.1 − 41.7i)31-s + ⋯
L(s)  = 1  + (0.822 + 0.568i)3-s + (0.439 − 0.898i)5-s + (0.402 − 1.50i)7-s + (0.352 + 0.935i)9-s + (−0.240 − 0.416i)11-s + (−0.200 − 0.746i)13-s + (0.872 − 0.488i)15-s + (0.373 + 0.373i)17-s + 1.50i·19-s + (1.18 − 1.00i)21-s + (0.454 + 1.69i)23-s + (−0.613 − 0.789i)25-s + (−0.242 + 0.970i)27-s + (−0.0788 + 0.0455i)29-s + (0.777 − 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.915 + 0.402i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ 0.915 + 0.402i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.04702 - 0.429916i\)
\(L(\frac12)\) \(\approx\) \(2.04702 - 0.429916i\)
\(L(2)\) 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 \( 1 + (-2.46 - 1.70i)T \)
5 \( 1 + (-2.19 + 4.49i)T \)
good7 \( 1 + (-2.81 + 10.5i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (2.64 + 4.57i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (2.60 + 9.70i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (-6.35 - 6.35i)T + 289iT^{2} \)
19 \( 1 - 28.5iT - 361T^{2} \)
23 \( 1 + (-10.4 - 38.9i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (2.28 - 1.32i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-24.1 + 41.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (14.8 + 14.8i)T + 1.36e3iT^{2} \)
41 \( 1 + (32.8 - 56.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-40.8 - 10.9i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (6.95 - 25.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (35.9 - 35.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (33.4 + 19.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (29.1 + 50.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-22.3 + 6.00i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 4.88T + 5.04e3T^{2} \)
73 \( 1 + (16.8 - 16.8i)T - 5.32e3iT^{2} \)
79 \( 1 + (79.7 - 46.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-139. - 37.3i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 10.7iT - 7.92e3T^{2} \)
97 \( 1 + (-9.66 + 36.0i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−12.66428411936269442956866201444, −11.12455199128304838226006770935, −10.14819209436468362057803553428, −9.538645679985918730010670483498, −8.082227372739895398049740970236, −7.72348665575184545890287780174, −5.72435177723999453924708398832, −4.53107320874349768447979002638, −3.45987463373122600911451186713, −1.39840605304019837635691892673, 2.13983564602408002090296326036, 2.89791256981206328414950632258, 4.94526416369292577374997547320, 6.43685289114755704434263940980, 7.20583277106765780671442196826, 8.618213590355528194678765310855, 9.203023898711238111838604293836, 10.44169466568377637411492656125, 11.74337714229896729245412592664, 12.44170348051363880370643208271

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