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Properties

Label 2-108-9.4-c3-0-1
Degree $2$
Conductor $108$
Sign $0.655 + 0.754i$
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  + (6.37 − 11.0i)5-s + (7.02 + 12.1i)7-s + (−21.2 − 36.8i)11-s + (36.2 − 62.7i)13-s + 59.6·17-s + 105.·19-s + (−0.112 + 0.195i)23-s + (−18.6 − 32.3i)25-s + (−112. − 195. i)29-s + (−100. + 174. i)31-s + 179.·35-s − 152.·37-s + (−244. + 424. i)41-s + (3.79 + 6.57i)43-s + (186. + 323. i)47-s + ⋯
L(s)  = 1  + (0.569 − 0.986i)5-s + (0.379 + 0.657i)7-s + (−0.583 − 1.01i)11-s + (0.772 − 1.33i)13-s + 0.850·17-s + 1.27·19-s + (−0.00102 + 0.00176i)23-s + (−0.149 − 0.258i)25-s + (−0.722 − 1.25i)29-s + (−0.582 + 1.00i)31-s + 0.864·35-s − 0.679·37-s + (−0.932 + 1.61i)41-s + (0.0134 + 0.0233i)43-s + (0.579 + 1.00i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.655 + 0.754i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ 0.655 + 0.754i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.60495 - 0.731556i\)
\(L(\frac12)\) \(\approx\) \(1.60495 - 0.731556i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-6.37 + 11.0i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-7.02 - 12.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (21.2 + 36.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-36.2 + 62.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 59.6T + 4.91e3T^{2} \)
19 \( 1 - 105.T + 6.85e3T^{2} \)
23 \( 1 + (0.112 - 0.195i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (112. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (100. - 174. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 152.T + 5.06e4T^{2} \)
41 \( 1 + (244. - 424. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-3.79 - 6.57i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-186. - 323. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 43.6T + 1.48e5T^{2} \)
59 \( 1 + (335. - 581. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (37.0 + 64.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (210. - 364. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 730.T + 3.57e5T^{2} \)
73 \( 1 - 473.T + 3.89e5T^{2} \)
79 \( 1 + (-264. - 458. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-13.0 - 22.6i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 415.T + 7.04e5T^{2} \)
97 \( 1 + (463. + 803. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.14153035229829597259143982924, −12.15359864213311094242685225230, −11.00006466291121156507867643897, −9.780917802161537685562290029172, −8.652897468187192026073441277333, −7.86887695385306704855140977479, −5.74365818668759514831955687625, −5.29545941711378763869726626505, −3.16690435528968200793987770574, −1.12322978972180077391822437651, 1.85921228953173603111288956228, 3.65371884616420700930501210720, 5.29664639856771086251717932999, 6.80310485693927456821688830641, 7.56331363203615970169740117166, 9.258011435687718671179992135486, 10.27227617643410162479026555347, 11.08986189468597051632275766481, 12.27317025320802554110111636331, 13.72088754083852358574208142592

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