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Properties

Label 2-108-4.3-c4-0-25
Degree $2$
Conductor $108$
Sign $-0.981 - 0.189i$
Analytic cond. $11.1639$
Root an. cond. $3.34125$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.54 − 3.08i)2-s + (−3.02 + 15.7i)4-s + 3.58·5-s + 16.2i·7-s + (56.1 − 30.6i)8-s + (−9.12 − 11.0i)10-s − 144. i·11-s − 223.·13-s + (50.1 − 41.4i)14-s + (−237. − 95.0i)16-s − 1.80·17-s − 70.1i·19-s + (−10.8 + 56.2i)20-s + (−446. + 368. i)22-s − 251. i·23-s + ⋯
L(s)  = 1  + (−0.636 − 0.771i)2-s + (−0.189 + 0.981i)4-s + 0.143·5-s + 0.331i·7-s + (0.877 − 0.479i)8-s + (−0.0912 − 0.110i)10-s − 1.19i·11-s − 1.32·13-s + (0.255 − 0.211i)14-s + (−0.928 − 0.371i)16-s − 0.00624·17-s − 0.194i·19-s + (−0.0271 + 0.140i)20-s + (−0.922 + 0.761i)22-s − 0.475i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.981 - 0.189i$
Analytic conductor: \(11.1639\)
Root analytic conductor: \(3.34125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :2),\ -0.981 - 0.189i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0314945 + 0.330051i\)
\(L(\frac12)\) \(\approx\) \(0.0314945 + 0.330051i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.54 + 3.08i)T \)
3 \( 1 \)
good5 \( 1 - 3.58T + 625T^{2} \)
7 \( 1 - 16.2iT - 2.40e3T^{2} \)
11 \( 1 + 144. iT - 1.46e4T^{2} \)
13 \( 1 + 223.T + 2.85e4T^{2} \)
17 \( 1 + 1.80T + 8.35e4T^{2} \)
19 \( 1 + 70.1iT - 1.30e5T^{2} \)
23 \( 1 + 251. iT - 2.79e5T^{2} \)
29 \( 1 + 1.64e3T + 7.07e5T^{2} \)
31 \( 1 - 1.04e3iT - 9.23e5T^{2} \)
37 \( 1 + 690.T + 1.87e6T^{2} \)
41 \( 1 + 993.T + 2.82e6T^{2} \)
43 \( 1 + 1.98e3iT - 3.41e6T^{2} \)
47 \( 1 + 3.19e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.86e3T + 7.89e6T^{2} \)
59 \( 1 + 5.57e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.83e3T + 1.38e7T^{2} \)
67 \( 1 - 5.28e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.87e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.52e3T + 2.83e7T^{2} \)
79 \( 1 + 7.36e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.00e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.14e4T + 6.27e7T^{2} \)
97 \( 1 + 9.02e3T + 8.85e7T^{2} \)
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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

−12.21886662162386002302096663926, −11.35625267897740550646988217767, −10.29033621695366271033463330184, −9.267912503939989345148939583029, −8.300722213171194149582264641748, −7.06379936316866683540687757958, −5.31959542051098464161563327452, −3.56783471375313475353289362255, −2.13564889093580539723554982946, −0.17217281424127234926720816060, 1.91884358153253504944754350470, 4.43977524991144986556668315748, 5.70644208594085004220538751470, 7.16592608621861003668812007620, 7.78043470788728756370121233791, 9.462412726198559041523567506605, 9.892651550873330731827608497450, 11.24072250137281206189410690492, 12.58830647569450341845830089412, 13.77013328020047740155996662814

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