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Properties

Label 2-108-4.3-c2-0-2
Degree $2$
Conductor $108$
Sign $-0.866 - 0.499i$
Analytic cond. $2.94278$
Root an. cond. $1.71545$
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  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s − 7·5-s + 8.66i·7-s − 7.99·8-s + (−7 − 12.1i)10-s + 8.66i·11-s + 20·13-s + (−15 + 8.66i)14-s + (−8 − 13.8i)16-s + 8·17-s − 10.3i·19-s + (13.9 − 24.2i)20-s + (−15 + 8.66i)22-s − 3.46i·23-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 1.40·5-s + 1.23i·7-s − 0.999·8-s + (−0.700 − 1.21i)10-s + 0.787i·11-s + 1.53·13-s + (−1.07 + 0.618i)14-s + (−0.5 − 0.866i)16-s + 0.470·17-s − 0.546i·19-s + (0.699 − 1.21i)20-s + (−0.681 + 0.393i)22-s − 0.150i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.866 - 0.499i$
Analytic conductor: \(2.94278\)
Root analytic conductor: \(1.71545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1),\ -0.866 - 0.499i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.305124 + 1.13873i\)
\(L(\frac12)\) \(\approx\) \(0.305124 + 1.13873i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 + 7T + 25T^{2} \)
7 \( 1 - 8.66iT - 49T^{2} \)
11 \( 1 - 8.66iT - 121T^{2} \)
13 \( 1 - 20T + 169T^{2} \)
17 \( 1 - 8T + 289T^{2} \)
19 \( 1 + 10.3iT - 361T^{2} \)
23 \( 1 + 3.46iT - 529T^{2} \)
29 \( 1 + 10T + 841T^{2} \)
31 \( 1 - 53.6iT - 961T^{2} \)
37 \( 1 + 10T + 1.36e3T^{2} \)
41 \( 1 - 50T + 1.68e3T^{2} \)
43 \( 1 + 17.3iT - 1.84e3T^{2} \)
47 \( 1 - 86.6iT - 2.20e3T^{2} \)
53 \( 1 - 47T + 2.80e3T^{2} \)
59 \( 1 + 34.6iT - 3.48e3T^{2} \)
61 \( 1 + 64T + 3.72e3T^{2} \)
67 \( 1 + 86.6iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 55T + 5.32e3T^{2} \)
79 \( 1 + 6.92iT - 6.24e3T^{2} \)
83 \( 1 - 29.4iT - 6.88e3T^{2} \)
89 \( 1 + 10T + 7.92e3T^{2} \)
97 \( 1 + 25T + 9.40e3T^{2} \)
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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

−14.11159480552858193627565991505, −12.70382142159322865412156595753, −12.12908326588426302492827968801, −11.10284807026164305307958769683, −9.089151567722652576589321718308, −8.318129157824399392637293886953, −7.25391532382315840977502948194, −5.94959904048751100675937442017, −4.58790222975333020474746093491, −3.28938179994876724343884246102, 0.801805983683099276174956392120, 3.57264637919876430099973986581, 4.09671298587819528734407371992, 5.94677577327664609242353531672, 7.57689051922074984065029873599, 8.683447153555869748696341533782, 10.27169228458798683966810714111, 11.14844941353113461753741700841, 11.74036510672811192097502507037, 13.09096465422954007633459593371

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