Properties

Label 2-108-36.11-c5-0-14
Degree $2$
Conductor $108$
Sign $0.231 + 0.972i$
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  + (0.0499 − 5.65i)2-s + (−31.9 − 0.564i)4-s + (−70.1 + 40.5i)5-s + (89.9 + 51.9i)7-s + (−4.79 + 180. i)8-s + (225. + 399. i)10-s + (214. − 371. i)11-s + (42.9 + 74.4i)13-s + (298. − 506. i)14-s + (1.02e3 + 36.1i)16-s − 1.50e3i·17-s − 1.22e3i·19-s + (2.26e3 − 1.25e3i)20-s + (−2.08e3 − 1.23e3i)22-s + (1.53e3 + 2.65e3i)23-s + ⋯
L(s)  = 1  + (0.00882 − 0.999i)2-s + (−0.999 − 0.0176i)4-s + (−1.25 + 0.724i)5-s + (0.694 + 0.400i)7-s + (−0.0264 + 0.999i)8-s + (0.713 + 1.26i)10-s + (0.533 − 0.924i)11-s + (0.0704 + 0.122i)13-s + (0.406 − 0.690i)14-s + (0.999 + 0.0352i)16-s − 1.26i·17-s − 0.778i·19-s + (1.26 − 0.702i)20-s + (−0.920 − 0.542i)22-s + (0.603 + 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(17.3214\)
Root analytic conductor: \(4.16190\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :5/2),\ 0.231 + 0.972i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.05204 - 0.831192i\)
\(L(\frac12)\) \(\approx\) \(1.05204 - 0.831192i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0499 + 5.65i)T \)
3 \( 1 \)
good5 \( 1 + (70.1 - 40.5i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-89.9 - 51.9i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-214. + 371. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-42.9 - 74.4i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.50e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.22e3iT - 2.47e6T^{2} \)
23 \( 1 + (-1.53e3 - 2.65e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-4.34e3 - 2.50e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (3.40e3 - 1.96e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 1.30e4T + 6.93e7T^{2} \)
41 \( 1 + (-1.50e4 + 8.70e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (5.75e3 + 3.32e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.00e4 + 1.74e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 1.80e4iT - 4.18e8T^{2} \)
59 \( 1 + (-1.47e4 - 2.55e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.19e4 - 2.07e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (348. - 201. i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.66e4T + 1.80e9T^{2} \)
73 \( 1 - 5.86e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.93e4 - 2.27e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (2.72e4 - 4.72e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 4.53e4iT - 5.58e9T^{2} \)
97 \( 1 + (-4.18e4 + 7.24e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.08969276652059218707446090067, −11.37997356773107300893165821524, −10.96030393665091132765267522436, −9.317093422325022237922242516478, −8.353154572164471133256472217282, −7.15453171177601121719147010682, −5.21801684286541337731526041104, −3.85987791565725182007031669292, −2.75003289459533216863829155584, −0.74116316361730629256267652956, 0.979223262590600048149489271066, 4.07563539460453322143966360740, 4.62407418643584791597311291317, 6.31099084063842298085149292120, 7.75118861987186080561425376241, 8.164991643738261372548882781672, 9.435924407268693313472194124761, 10.88247190937492654162910461075, 12.26844815264055969427297081560, 12.87306078674304402287068916572

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