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Properties

Label 2-108-12.11-c5-0-12
Degree $2$
Conductor $108$
Sign $0.970 - 0.241i$
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  + (−4.45 − 3.48i)2-s + (7.72 + 31.0i)4-s + 40.4i·5-s − 148. i·7-s + (73.7 − 165. i)8-s + (141. − 180. i)10-s − 608.·11-s + 883.·13-s + (−517. + 661. i)14-s + (−904. + 479. i)16-s + 1.13e3i·17-s + 1.11e3i·19-s + (−1.25e3 + 312. i)20-s + (2.71e3 + 2.11e3i)22-s + 1.34e3·23-s + ⋯
L(s)  = 1  + (−0.787 − 0.615i)2-s + (0.241 + 0.970i)4-s + 0.724i·5-s − 1.14i·7-s + (0.407 − 0.913i)8-s + (0.446 − 0.570i)10-s − 1.51·11-s + 1.45·13-s + (−0.705 + 0.902i)14-s + (−0.883 + 0.468i)16-s + 0.951i·17-s + 0.706i·19-s + (−0.702 + 0.174i)20-s + (1.19 + 0.933i)22-s + 0.530·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.970 - 0.241i$
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: \((2,\ 108,\ (\ :5/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.09605 + 0.134233i\)
\(L(\frac12)\) \(\approx\) \(1.09605 + 0.134233i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.45 + 3.48i)T \)
3 \( 1 \)
good5 \( 1 - 40.4iT - 3.12e3T^{2} \)
7 \( 1 + 148. iT - 1.68e4T^{2} \)
11 \( 1 + 608.T + 1.61e5T^{2} \)
13 \( 1 - 883.T + 3.71e5T^{2} \)
17 \( 1 - 1.13e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.11e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.34e3T + 6.43e6T^{2} \)
29 \( 1 + 1.85e3iT - 2.05e7T^{2} \)
31 \( 1 + 606. iT - 2.86e7T^{2} \)
37 \( 1 - 1.36e4T + 6.93e7T^{2} \)
41 \( 1 - 2.12e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.24e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.62e4T + 2.29e8T^{2} \)
53 \( 1 - 1.34e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.12e4T + 7.14e8T^{2} \)
61 \( 1 - 6.50e3T + 8.44e8T^{2} \)
67 \( 1 - 2.35e3iT - 1.35e9T^{2} \)
71 \( 1 + 2.69e4T + 1.80e9T^{2} \)
73 \( 1 + 5.23e4T + 2.07e9T^{2} \)
79 \( 1 - 3.57e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.29e4T + 3.93e9T^{2} \)
89 \( 1 - 5.45e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.42e5T + 8.58e9T^{2} \)
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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

−12.92874384679686586060003148178, −11.21840676646599226555988204483, −10.69738152248627283619207968073, −9.951086603495229514960086413366, −8.330875306765483024165421654173, −7.57367760532358049501948381616, −6.28253445427515889136558011437, −4.08369416305890113663602020835, −2.86692095914772089440202392993, −1.08120847582640209705496781823, 0.68096574735287298065108836409, 2.48737282816977234546048265287, 5.03715617007176707004992508438, 5.80382433758882510689025738054, 7.32989632194224092581430726359, 8.639474264230656341477987578605, 9.007183643385948239557574401306, 10.47118086079618204545746575021, 11.45901546518674664955679600304, 12.81510987557134419838726704378

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