Properties

Label 2-108-12.11-c5-0-0
Degree $2$
Conductor $108$
Sign $0.287 - 0.957i$
Analytic cond. $17.3214$
Root an. cond. $4.16190$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.821 − 5.59i)2-s + (−30.6 − 9.19i)4-s − 8.15i·5-s − 63.0i·7-s + (−76.6 + 164. i)8-s + (−45.6 − 6.69i)10-s − 442.·11-s − 76.3·13-s + (−352. − 51.7i)14-s + (855. + 563. i)16-s + 643. i·17-s + 2.18e3i·19-s + (−74.9 + 250. i)20-s + (−363. + 2.47e3i)22-s − 2.73e3·23-s + ⋯
L(s)  = 1  + (0.145 − 0.989i)2-s + (−0.957 − 0.287i)4-s − 0.145i·5-s − 0.485i·7-s + (−0.423 + 0.906i)8-s + (−0.144 − 0.0211i)10-s − 1.10·11-s − 0.125·13-s + (−0.480 − 0.0705i)14-s + (0.835 + 0.550i)16-s + 0.540i·17-s + 1.39i·19-s + (−0.0419 + 0.139i)20-s + (−0.160 + 1.09i)22-s − 1.07·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.287 - 0.957i$
Analytic conductor: \(17.3214\)
Root analytic conductor: \(4.16190\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :5/2),\ 0.287 - 0.957i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.291469 + 0.216891i\)
\(L(\frac12)\) \(\approx\) \(0.291469 + 0.216891i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.821 + 5.59i)T \)
3 \( 1 \)
good5 \( 1 + 8.15iT - 3.12e3T^{2} \)
7 \( 1 + 63.0iT - 1.68e4T^{2} \)
11 \( 1 + 442.T + 1.61e5T^{2} \)
13 \( 1 + 76.3T + 3.71e5T^{2} \)
17 \( 1 - 643. iT - 1.41e6T^{2} \)
19 \( 1 - 2.18e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.73e3T + 6.43e6T^{2} \)
29 \( 1 - 1.50e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.50e3iT - 2.86e7T^{2} \)
37 \( 1 + 4.82e3T + 6.93e7T^{2} \)
41 \( 1 + 1.09e4iT - 1.15e8T^{2} \)
43 \( 1 - 8.35e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.39e4T + 2.29e8T^{2} \)
53 \( 1 - 2.28e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.83e4T + 7.14e8T^{2} \)
61 \( 1 - 5.10e3T + 8.44e8T^{2} \)
67 \( 1 + 3.73e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.50e4T + 1.80e9T^{2} \)
73 \( 1 + 6.52e4T + 2.07e9T^{2} \)
79 \( 1 + 7.32e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.52e4T + 3.93e9T^{2} \)
89 \( 1 - 9.20e3iT - 5.58e9T^{2} \)
97 \( 1 - 1.25e5T + 8.58e9T^{2} \)
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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

−12.77289003137173566962245761174, −12.08115575269090284596758989672, −10.61459075301398949434673082548, −10.23938149051712404902894630470, −8.782572172909747557521524072168, −7.71091522652835688910186813090, −5.82924293301247829492002238329, −4.55814994966356496398061287484, −3.21720840408391944515033280059, −1.60304599912431397563386432224, 0.13036777292092968409116174284, 2.79994953648595759278393695543, 4.61388191229714509083594842881, 5.66791644397259407882318292826, 6.93967237991142753612565572291, 8.004046209897939829077621753315, 9.069014215232798261700608450049, 10.19510855525732366113680085231, 11.64324948022896362440241330150, 12.88298058723591011946992996080

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