Properties

Label 2-108-12.11-c3-0-14
Degree $2$
Conductor $108$
Sign $0.525 + 0.850i$
Analytic cond. $6.37220$
Root an. cond. $2.52432$
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  + (−2.72 + 0.772i)2-s + (6.80 − 4.20i)4-s + 3.33i·5-s − 16.9i·7-s + (−15.2 + 16.6i)8-s + (−2.57 − 9.06i)10-s − 16.8·11-s + 25.0·13-s + (13.0 + 45.9i)14-s + (28.6 − 57.2i)16-s − 116. i·17-s − 85.4i·19-s + (14.0 + 22.6i)20-s + (45.7 − 13.0i)22-s + 158.·23-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)2-s + (0.850 − 0.525i)4-s + 0.297i·5-s − 0.912i·7-s + (−0.674 + 0.737i)8-s + (−0.0813 − 0.286i)10-s − 0.461·11-s + 0.535·13-s + (0.249 + 0.877i)14-s + (0.447 − 0.894i)16-s − 1.66i·17-s − 1.03i·19-s + (0.156 + 0.253i)20-s + (0.443 − 0.126i)22-s + 1.43·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.782881 - 0.436617i\)
\(L(\frac12)\) \(\approx\) \(0.782881 - 0.436617i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.72 - 0.772i)T \)
3 \( 1 \)
good5 \( 1 - 3.33iT - 125T^{2} \)
7 \( 1 + 16.9iT - 343T^{2} \)
11 \( 1 + 16.8T + 1.33e3T^{2} \)
13 \( 1 - 25.0T + 2.19e3T^{2} \)
17 \( 1 + 116. iT - 4.91e3T^{2} \)
19 \( 1 + 85.4iT - 6.85e3T^{2} \)
23 \( 1 - 158.T + 1.21e4T^{2} \)
29 \( 1 + 269. iT - 2.43e4T^{2} \)
31 \( 1 - 36.0iT - 2.97e4T^{2} \)
37 \( 1 + 353.T + 5.06e4T^{2} \)
41 \( 1 + 144. iT - 6.89e4T^{2} \)
43 \( 1 - 368. iT - 7.95e4T^{2} \)
47 \( 1 + 397.T + 1.03e5T^{2} \)
53 \( 1 - 96.0iT - 1.48e5T^{2} \)
59 \( 1 - 294.T + 2.05e5T^{2} \)
61 \( 1 - 146.T + 2.26e5T^{2} \)
67 \( 1 - 301. iT - 3.00e5T^{2} \)
71 \( 1 - 687.T + 3.57e5T^{2} \)
73 \( 1 + 312.T + 3.89e5T^{2} \)
79 \( 1 + 602. iT - 4.93e5T^{2} \)
83 \( 1 + 1.33e3T + 5.71e5T^{2} \)
89 \( 1 - 856. iT - 7.04e5T^{2} \)
97 \( 1 - 218.T + 9.12e5T^{2} \)
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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

−13.21194992123577017025230816826, −11.53223524118579545405754092814, −10.84396794327313272470286371617, −9.808986567833794513515697316799, −8.735378335856488336128252027957, −7.41011055341046585734367917318, −6.72590802678130999495595801274, −5.02553571407447225969326975812, −2.86202390242230195024421659298, −0.70683793051078551320981593027, 1.61712412330441366289475659958, 3.34629416603008498227498878650, 5.50130964753293465178669971259, 6.83604676861352659327123452745, 8.382359359986771037012460607086, 8.826080840488870352759260005454, 10.24993933729060766147362556168, 11.06142891064106441812153610087, 12.37808413173628357637475485060, 12.88173156146921259005210082016

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