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Properties

Label 2-180-9.7-c3-0-10
Degree $2$
Conductor $180$
Sign $-0.904 + 0.425i$
Analytic cond. $10.6203$
Root an. cond. $3.25888$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.39 − 4.61i)3-s + (−2.5 − 4.33i)5-s + (−12.3 + 21.3i)7-s + (−15.5 − 22.0i)9-s + (31.3 − 54.2i)11-s + (−29.8 − 51.6i)13-s + (−25.9 + 1.17i)15-s − 85.2·17-s − 75.1·19-s + (68.8 + 107. i)21-s + (48.7 + 84.4i)23-s + (−12.5 + 21.6i)25-s + (−139. + 18.9i)27-s + (−79.6 + 137. i)29-s + (−159. − 275. i)31-s + ⋯
L(s)  = 1  + (0.460 − 0.887i)3-s + (−0.223 − 0.387i)5-s + (−0.664 + 1.15i)7-s + (−0.576 − 0.817i)9-s + (0.858 − 1.48i)11-s + (−0.636 − 1.10i)13-s + (−0.446 + 0.0202i)15-s − 1.21·17-s − 0.907·19-s + (0.715 + 1.11i)21-s + (0.442 + 0.765i)23-s + (−0.100 + 0.173i)25-s + (−0.990 + 0.135i)27-s + (−0.509 + 0.883i)29-s + (−0.922 − 1.59i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.904 + 0.425i$
Analytic conductor: \(10.6203\)
Root analytic conductor: \(3.25888\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :3/2),\ -0.904 + 0.425i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.234612 - 1.05049i\)
\(L(\frac12)\) \(\approx\) \(0.234612 - 1.05049i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.39 + 4.61i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
good7 \( 1 + (12.3 - 21.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-31.3 + 54.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (29.8 + 51.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 85.2T + 4.91e3T^{2} \)
19 \( 1 + 75.1T + 6.85e3T^{2} \)
23 \( 1 + (-48.7 - 84.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (79.6 - 137. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (159. + 275. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 393.T + 5.06e4T^{2} \)
41 \( 1 + (42.5 + 73.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-20.3 + 35.3i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-45.1 + 78.1i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 691.T + 1.48e5T^{2} \)
59 \( 1 + (141. + 245. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-216. + 374. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (154. + 267. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 302.T + 3.57e5T^{2} \)
73 \( 1 + 457.T + 3.89e5T^{2} \)
79 \( 1 + (-420. + 728. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-394. + 682. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.11e3T + 7.04e5T^{2} \)
97 \( 1 + (598. - 1.03e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.92387850118314250990146376955, −11.08853560409944250106265320338, −9.246210136198462713250589886507, −8.836789195075785681407061025625, −7.76116966145161631903099260399, −6.40378339830971607444535506410, −5.62062872431967898637191015956, −3.57945833468572480716799167645, −2.36010488699218788467671208265, −0.43350627485104046328925229200, 2.31601217197303814851681593180, 4.07135969824889579634245090976, 4.45073135639204254619881244852, 6.64867767202237295427544139641, 7.27545632149358962781408975490, 8.873031410952193217482627343731, 9.685232806270943785832696684961, 10.47909572304930065977314292688, 11.42291278471251836562041033162, 12.70009624965334070806371631904

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