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Properties

Label 2-540-180.119-c1-0-30
Degree $2$
Conductor $540$
Sign $0.203 + 0.978i$
Analytic cond. $4.31192$
Root an. cond. $2.07651$
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  + (1.24 − 0.668i)2-s + (1.10 − 1.66i)4-s + (1.69 − 1.46i)5-s + (0.667 − 1.15i)7-s + (0.265 − 2.81i)8-s + (1.13 − 2.95i)10-s + (−2.18 + 3.78i)11-s + (3.56 − 2.05i)13-s + (0.0591 − 1.88i)14-s + (−1.55 − 3.68i)16-s − 6.45·17-s + 5.84i·19-s + (−0.562 − 4.43i)20-s + (−0.194 + 6.18i)22-s + (−0.0875 + 0.0505i)23-s + ⋯
L(s)  = 1  + (0.881 − 0.472i)2-s + (0.553 − 0.832i)4-s + (0.756 − 0.653i)5-s + (0.252 − 0.436i)7-s + (0.0939 − 0.995i)8-s + (0.358 − 0.933i)10-s + (−0.659 + 1.14i)11-s + (0.987 − 0.570i)13-s + (0.0158 − 0.504i)14-s + (−0.387 − 0.921i)16-s − 1.56·17-s + 1.34i·19-s + (−0.125 − 0.992i)20-s + (−0.0413 + 1.31i)22-s + (−0.0182 + 0.0105i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.203 + 0.978i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.203 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07495 - 1.68716i\)
\(L(\frac12)\) \(\approx\) \(2.07495 - 1.68716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 + 0.668i)T \)
3 \( 1 \)
5 \( 1 + (-1.69 + 1.46i)T \)
good7 \( 1 + (-0.667 + 1.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.56 + 2.05i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.45T + 17T^{2} \)
19 \( 1 - 5.84iT - 19T^{2} \)
23 \( 1 + (0.0875 - 0.0505i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.53 - 2.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.24iT - 37T^{2} \)
41 \( 1 + (3.50 - 2.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.92 - 3.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.00 - 1.73i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.77T + 53T^{2} \)
59 \( 1 + (-1.37 - 2.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.216 + 0.374i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 - 7.28iT - 73T^{2} \)
79 \( 1 + (-2.58 - 1.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.6 + 6.70i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.97iT - 89T^{2} \)
97 \( 1 + (-2.08 - 1.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.48212157818389229186008744899, −10.16870355868000546703854487203, −9.016236978740566224379942413839, −7.919043562881284969390632160786, −6.67799010530715631335252836884, −5.82043221402303128747759235955, −4.83322939307464485429538181164, −4.08304405639379613501784220473, −2.51015314560668930002668367875, −1.38562159009247686764800855876, 2.24223225908006317692316702634, 3.13884807019227767040249777466, 4.52069912112870285121779007395, 5.55147744775417226742714306224, 6.39219026503918916620564433804, 6.98011488475634568290893293787, 8.462714566987672288838154327715, 8.861647431914519252294674672739, 10.43831849874904778822154317824, 11.18566505865553975630404453988

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