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Properties

Label 2-108-4.3-c2-0-12
Degree $2$
Conductor $108$
Sign $-0.267 + 0.963i$
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  + (0.270 − 1.98i)2-s + (−3.85 − 1.07i)4-s + 7.40·5-s − 9.47i·7-s + (−3.16 + 7.34i)8-s + (2.00 − 14.6i)10-s − 6.77i·11-s − 14.4·13-s + (−18.7 − 2.55i)14-s + (13.7 + 8.25i)16-s + 17.8·17-s + 5.19i·19-s + (−28.5 − 7.92i)20-s + (−13.4 − 1.82i)22-s + 24.9i·23-s + ⋯
L(s)  = 1  + (0.135 − 0.990i)2-s + (−0.963 − 0.267i)4-s + 1.48·5-s − 1.35i·7-s + (−0.395 + 0.918i)8-s + (0.200 − 1.46i)10-s − 0.615i·11-s − 1.10·13-s + (−1.34 − 0.182i)14-s + (0.856 + 0.515i)16-s + 1.05·17-s + 0.273i·19-s + (−1.42 − 0.396i)20-s + (−0.609 − 0.0831i)22-s + 1.08i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.267 + 0.963i$
Analytic conductor: \(2.94278\)
Root analytic conductor: \(1.71545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1),\ -0.267 + 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.940165 - 1.23688i\)
\(L(\frac12)\) \(\approx\) \(0.940165 - 1.23688i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.270 + 1.98i)T \)
3 \( 1 \)
good5 \( 1 - 7.40T + 25T^{2} \)
7 \( 1 + 9.47iT - 49T^{2} \)
11 \( 1 + 6.77iT - 121T^{2} \)
13 \( 1 + 14.4T + 169T^{2} \)
17 \( 1 - 17.8T + 289T^{2} \)
19 \( 1 - 5.19iT - 361T^{2} \)
23 \( 1 - 24.9iT - 529T^{2} \)
29 \( 1 - 29.6T + 841T^{2} \)
31 \( 1 - 17.1iT - 961T^{2} \)
37 \( 1 - 6.41T + 1.36e3T^{2} \)
41 \( 1 - 8.64T + 1.68e3T^{2} \)
43 \( 1 - 50.1iT - 1.84e3T^{2} \)
47 \( 1 - 52.0iT - 2.20e3T^{2} \)
53 \( 1 + 82.6T + 2.80e3T^{2} \)
59 \( 1 - 83.7iT - 3.48e3T^{2} \)
61 \( 1 + 8.41T + 3.72e3T^{2} \)
67 \( 1 + 29.0iT - 4.48e3T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 - 2.16T + 5.32e3T^{2} \)
79 \( 1 + 149. iT - 6.24e3T^{2} \)
83 \( 1 - 13.5iT - 6.88e3T^{2} \)
89 \( 1 - 17.8T + 7.92e3T^{2} \)
97 \( 1 + 138.T + 9.40e3T^{2} \)
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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

−13.26964367392940585254326331458, −12.18979866569002875261666570597, −10.81664035832323082902888711111, −10.05030246877607827334558334389, −9.399045187058201661497527939746, −7.72777644333332563005023527614, −6.02973085528930368936697458476, −4.78679579013376341227381725630, −3.10074580534916915835852164597, −1.32552692312873136290682505424, 2.47686259205557726263471131310, 4.95919035609663264046486485897, 5.75620782509359090412196574117, 6.83547351274826831008980645917, 8.345706183757296877959113429434, 9.458944098851275205878234160332, 10.03350843837695944632395516015, 12.18951909533009392881659070944, 12.79048341116104596157080139475, 14.07468925980341068210050078996

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