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Properties

Label 2-180-12.11-c7-0-0
Degree $2$
Conductor $180$
Sign $-0.279 + 0.960i$
Analytic cond. $56.2293$
Root an. cond. $7.49862$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.21 + 6.56i)2-s + (41.7 + 121. i)4-s + 125i·5-s − 19.8i·7-s + (−410. + 1.38e3i)8-s + (−821. + 1.15e3i)10-s − 6.77e3·11-s + 974.·13-s + (130. − 182. i)14-s + (−1.29e4 + 1.00e4i)16-s − 1.43e4i·17-s − 1.91e4i·19-s + (−1.51e4 + 5.21e3i)20-s + (−6.24e4 − 4.45e4i)22-s − 1.12e4·23-s + ⋯
L(s)  = 1  + (0.814 + 0.580i)2-s + (0.325 + 0.945i)4-s + 0.447i·5-s − 0.0218i·7-s + (−0.283 + 0.958i)8-s + (−0.259 + 0.364i)10-s − 1.53·11-s + 0.122·13-s + (0.0126 − 0.0177i)14-s + (−0.787 + 0.616i)16-s − 0.710i·17-s − 0.640i·19-s + (−0.422 + 0.145i)20-s + (−1.25 − 0.891i)22-s − 0.192·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.279 + 0.960i$
Analytic conductor: \(56.2293\)
Root analytic conductor: \(7.49862\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :7/2),\ -0.279 + 0.960i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.05876505465\)
\(L(\frac12)\) \(\approx\) \(0.05876505465\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.21 - 6.56i)T \)
3 \( 1 \)
5 \( 1 - 125iT \)
good7 \( 1 + 19.8iT - 8.23e5T^{2} \)
11 \( 1 + 6.77e3T + 1.94e7T^{2} \)
13 \( 1 - 974.T + 6.27e7T^{2} \)
17 \( 1 + 1.43e4iT - 4.10e8T^{2} \)
19 \( 1 + 1.91e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.12e4T + 3.40e9T^{2} \)
29 \( 1 + 6.18e3iT - 1.72e10T^{2} \)
31 \( 1 - 5.30e4iT - 2.75e10T^{2} \)
37 \( 1 + 3.87e5T + 9.49e10T^{2} \)
41 \( 1 + 3.84e5iT - 1.94e11T^{2} \)
43 \( 1 - 3.21e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.26e5T + 5.06e11T^{2} \)
53 \( 1 + 1.45e6iT - 1.17e12T^{2} \)
59 \( 1 + 9.33e5T + 2.48e12T^{2} \)
61 \( 1 - 8.74e5T + 3.14e12T^{2} \)
67 \( 1 + 9.11e5iT - 6.06e12T^{2} \)
71 \( 1 + 3.95e6T + 9.09e12T^{2} \)
73 \( 1 + 2.99e6T + 1.10e13T^{2} \)
79 \( 1 + 6.37e6iT - 1.92e13T^{2} \)
83 \( 1 + 4.82e6T + 2.71e13T^{2} \)
89 \( 1 - 1.15e7iT - 4.42e13T^{2} \)
97 \( 1 - 9.68e6T + 8.07e13T^{2} \)
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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

−12.22400529618688824228888420857, −11.19977138855124119749551432667, −10.27346440037126932039159787419, −8.773301915744193625772195666838, −7.68567410054987569257851248237, −6.90261333770538537248392013396, −5.64307776803364427595633369504, −4.76095612450070521052943771096, −3.30940115712706426229266135393, −2.33544842322502096747678153632, 0.01024156432908233272112778874, 1.51577711319821764839049783456, 2.74894586961268136210484427265, 4.03857923683159447857003559127, 5.18882067876576718312696121884, 6.01444305128964664755352532654, 7.50278390233215697268629554239, 8.694501024247929964075153588582, 10.06890321083055286213174742608, 10.65474256655022724533721160900

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