Properties

Label 2-2112-33.32-c1-0-89
Degree $2$
Conductor $2112$
Sign $-0.957 - 0.288i$
Analytic cond. $16.8644$
Root an. cond. $4.10662$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 1.65i)3-s − 3.31i·5-s + (−2.5 − 1.65i)9-s − 3.31i·11-s + (−5.5 − 1.65i)15-s − 3.31i·23-s − 6·25-s + (−4 + 3.31i)27-s − 5·31-s + (−5.5 − 1.65i)33-s + 7·37-s + (−5.5 + 8.29i)45-s + 6.63i·47-s + 7·49-s − 13.2i·53-s + ⋯
L(s)  = 1  + (0.288 − 0.957i)3-s − 1.48i·5-s + (−0.833 − 0.552i)9-s − 1.00i·11-s + (−1.42 − 0.428i)15-s − 0.691i·23-s − 1.20·25-s + (−0.769 + 0.638i)27-s − 0.898·31-s + (−0.957 − 0.288i)33-s + 1.15·37-s + (−0.819 + 1.23i)45-s + 0.967i·47-s + 49-s − 1.82i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $-0.957 - 0.288i$
Analytic conductor: \(16.8644\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ -0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.416826841\)
\(L(\frac12)\) \(\approx\) \(1.416826841\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 1.65i)T \)
11 \( 1 + 3.31iT \)
good5 \( 1 + 3.31iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 3.31iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 + 13.2iT - 53T^{2} \)
59 \( 1 - 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.735542653030680697902887100002, −8.020554353859043114915957538936, −7.27844567579692207367038681363, −6.18863176910079900482594720154, −5.63725309402560693470137501505, −4.68881497114882490288591203939, −3.66593393715245845962357175117, −2.53470721568946959113476013715, −1.36535754743472841751745351922, −0.48464551870805226415211461413, 2.06134838773840352620194629119, 2.91092074279684780401366957678, 3.71491036660840051082780269267, 4.53708104967998504886270956656, 5.55411432761907979350050358359, 6.38128631614240847576100861824, 7.34503481884025642830063362000, 7.77567340058232651941051968816, 9.020535395406758595907800001202, 9.549713229132018522012750565200

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