↑

Properties

Label 2-2112-24.11-c1-0-73
Degree $2$
Conductor $2112$
Sign $-0.938 + 0.346i$
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  + (−1.41 + i)3-s + 2·5-s + 0.449i·7-s + (1.00 − 2.82i)9-s − i·11-s − 3.46i·13-s + (−2.82 + 2i)15-s − 2.04i·17-s − 7.70·19-s + (−0.449 − 0.635i)21-s − 0.635·23-s − 25-s + (1.41 + 5.00i)27-s − 9.34·29-s + 2.89i·31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + 0.894·5-s + 0.169i·7-s + (0.333 − 0.942i)9-s − 0.301i·11-s − 0.960i·13-s + (−0.730 + 0.516i)15-s − 0.497i·17-s − 1.76·19-s + (−0.0980 − 0.138i)21-s − 0.132·23-s − 0.200·25-s + (0.272 + 0.962i)27-s − 1.73·29-s + 0.520i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $-0.938 + 0.346i$
Analytic conductor: \(16.8644\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ -0.938 + 0.346i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05870761400\)
\(L(\frac12)\) \(\approx\) \(0.05870761400\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
11 \( 1 + iT \)
good5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 0.449iT - 7T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 2.04iT - 17T^{2} \)
19 \( 1 + 7.70T + 19T^{2} \)
23 \( 1 + 0.635T + 23T^{2} \)
29 \( 1 + 9.34T + 29T^{2} \)
31 \( 1 - 2.89iT - 31T^{2} \)
37 \( 1 - 9.12iT - 37T^{2} \)
41 \( 1 + 0.778iT - 41T^{2} \)
43 \( 1 + 4.87T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 6.89T + 53T^{2} \)
59 \( 1 - 9.79iT - 59T^{2} \)
61 \( 1 + 2.19iT - 61T^{2} \)
67 \( 1 + 4.09T + 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + 2.89T + 73T^{2} \)
79 \( 1 + 5.34iT - 79T^{2} \)
83 \( 1 - 9.79iT - 83T^{2} \)
89 \( 1 - 13.8iT - 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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

−8.942721207735521173303544161034, −8.085691753970244064988114339821, −6.95514095660109968780862910778, −6.15525952770751630740453267232, −5.63989811977158387092715045026, −4.88862706767651440982502489783, −3.91422966473477744134249838517, −2.84324942980960665980983648465, −1.59104106857047945633466112636, −0.02120086238198046812986984360, 1.78535843240288460388494133151, 2.11231206781878870661131921048, 3.91361350548809778936838609222, 4.68784444794374693328413738510, 5.77579379695175694375963586757, 6.17937601308171528066160776872, 6.97986561844950169942843253880, 7.71207193854566948994642699160, 8.717649959456335555707893317317, 9.496561721012174096565870910616

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