Properties

Degree 1
Conductor $ 2^{2} \cdot 11 $
Sign $-0.935 - 0.352i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(44\)    =    \(2^{2} \cdot 11\)
\( \varepsilon \)  =  $-0.935 - 0.352i$
motivic weight  =  \(0\)
character  :  $\chi_{44} (31, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 44,\ (1:\ ),\ -0.935 - 0.352i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.03678506706 + 0.2021961541i$
$L(\frac12,\chi)$  $\approx$  $-0.03678506706 + 0.2021961541i$
$L(\chi,1)$  $\approx$  0.5479073433 + 0.2061831488i
$L(1,\chi)$  $\approx$  0.5479073433 + 0.2061831488i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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