Properties

Label 2-108-108.31-c2-0-1
Degree $2$
Conductor $108$
Sign $-0.715 + 0.698i$
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  + (0.277 + 1.98i)2-s + (−2.64 + 1.42i)3-s + (−3.84 + 1.09i)4-s + (−0.542 + 3.07i)5-s + (−3.55 − 4.83i)6-s + (−2.98 − 3.56i)7-s + (−3.24 − 7.31i)8-s + (4.94 − 7.51i)9-s + (−6.24 − 0.220i)10-s + (2.54 − 0.448i)11-s + (8.59 − 8.37i)12-s + (−16.7 + 6.09i)13-s + (6.22 − 6.90i)14-s + (−2.94 − 8.89i)15-s + (13.5 − 8.45i)16-s + (−10.3 + 17.9i)17-s + ⋯
L(s)  = 1  + (0.138 + 0.990i)2-s + (−0.880 + 0.474i)3-s + (−0.961 + 0.274i)4-s + (−0.108 + 0.615i)5-s + (−0.592 − 0.805i)6-s + (−0.426 − 0.508i)7-s + (−0.405 − 0.914i)8-s + (0.549 − 0.835i)9-s + (−0.624 − 0.0220i)10-s + (0.231 − 0.0408i)11-s + (0.715 − 0.698i)12-s + (−1.28 + 0.468i)13-s + (0.444 − 0.493i)14-s + (−0.196 − 0.592i)15-s + (0.848 − 0.528i)16-s + (−0.609 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.715 + 0.698i$
Analytic conductor: \(2.94278\)
Root analytic conductor: \(1.71545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1),\ -0.715 + 0.698i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.159530 - 0.392047i\)
\(L(\frac12)\) \(\approx\) \(0.159530 - 0.392047i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.277 - 1.98i)T \)
3 \( 1 + (2.64 - 1.42i)T \)
good5 \( 1 + (0.542 - 3.07i)T + (-23.4 - 8.55i)T^{2} \)
7 \( 1 + (2.98 + 3.56i)T + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (-2.54 + 0.448i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (16.7 - 6.09i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (10.3 - 17.9i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (24.5 - 14.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.6 + 17.4i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-5.90 - 2.14i)T + (644. + 540. i)T^{2} \)
31 \( 1 + (22.3 - 26.6i)T + (-166. - 946. i)T^{2} \)
37 \( 1 + (22.9 - 39.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (4.27 - 1.55i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (12.6 - 2.23i)T + (1.73e3 - 632. i)T^{2} \)
47 \( 1 + (-1.37 - 1.64i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 31.8T + 2.80e3T^{2} \)
59 \( 1 + (-111. - 19.6i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-63.3 + 53.1i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (-10.8 - 29.8i)T + (-3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (0.0648 + 0.0374i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (60.1 + 104. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (0.117 - 0.323i)T + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (32.5 - 89.3i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-59.4 - 102. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-11.2 - 63.5i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−14.64236747293037623248277065982, −13.04106598686919119495437881703, −12.23363683910630498614574861538, −10.72857686638980581954762573274, −9.964035562945888826430740124558, −8.637263125781377888000244162234, −6.93976576061417354388761437017, −6.49965047703463905015028236556, −4.94305626796337063751461149720, −3.80213756955187013705589037608, 0.32315566498902076177870235657, 2.39818189267944063419009908780, 4.58014100876519937472505519214, 5.50309674713277386825297616283, 7.10380500269403468230898519048, 8.748656864546878763524105488120, 9.754610876076640011493653314039, 10.98680752663204071019685467578, 11.86118149268767849192499522427, 12.72489357664392322966698697991

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