↑

Properties

Label 1-44-44.27-r1-0-0
Degree $1$
Conductor $44$
Sign $-0.935 + 0.352i$
Analytic cond. $4.72845$
Root an. cond. $4.72845$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + 21-s − 23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)35-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + 21-s − 23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $-0.935 + 0.352i$
Analytic conductor: \(4.72845\)
Root analytic conductor: \(4.72845\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 44,\ (1:\ ),\ -0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03678506706 - 0.2021961541i\)
\(L(\frac12)\) \(\approx\) \(-0.03678506706 - 0.2021961541i\)
\(L(1)\) \(\approx\) \(0.5479073433 - 0.2061831488i\)
\(L(1)\) \(\approx\) \(0.5479073433 - 0.2061831488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.809 + 0.587i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−34.63561271346805585685758397276, −33.70548465028647270089104909642, −32.584620101713475131929444325135, −31.55146364280755424391898237197, −30.16306438955331550861899947473, −28.98156038300337285510602249552, −27.49199485996561679282736524317, −26.84088196274220243382913500941, −25.84373911290320838637159666876, −23.86940696051006425879827547350, −22.82694844042733137259402718025, −21.987413864811938353207944003740, −20.37466811850692118234705027316, −19.49700145616714973162167821585, −17.69171397640217301217099224976, −16.47239629567381648170827782753, −15.36244957967819663789763698516, −14.24514723275808642433370913896, −12.283490172599583076923866905989, −10.83301001851822001849206789943, −10.03316551602902323470123474706, −8.09705230451703433549964021748, −6.496789264594559951946487879824, −4.525598070353275168379219736539, −3.34154063195114266179661784631, 0.12371587598200842482254348227, 2.38490350203062896603436883931, 4.77537411469322920743218112564, 6.4327188802524382237351277976, 7.874409491957865883165014474005, 9.17132988297724527647009334580, 11.50028823646953568410290490130, 12.22321464224830853601482580265, 13.45829820681218960220081735530, 15.2365151486284314006385652574, 16.504500915881277138307712666562, 17.882095224734068308326679209, 19.142741832837080971014893962348, 19.92794371134630369674614163971, 21.79252406110680404451420081170, 23.01075097004080237393764053177, 24.23315033133611475616689250457, 24.86767825570663312196012196497, 26.4757667302328610479353990248, 28.1022201174370869622603656521, 28.69655319127780551263673170031, 30.09396645593261430820110630713, 31.337337202868234907494519780297, 31.971372261854370435741959114441, 33.92060258104603917863875901466

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