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Properties

Label 1-1441-1441.439-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.986 + 0.163i$
Analytic cond. $154.856$
Root an. cond. $154.856$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.399 − 0.916i)2-s + (−0.262 + 0.964i)3-s + (−0.681 + 0.732i)4-s + (−0.998 + 0.0483i)5-s + (0.989 − 0.144i)6-s + (0.168 + 0.985i)7-s + (0.943 + 0.331i)8-s + (−0.861 − 0.506i)9-s + (0.443 + 0.896i)10-s + (−0.527 − 0.849i)12-s + (0.443 − 0.896i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (−0.0724 − 0.997i)16-s + (−0.926 + 0.377i)17-s + (−0.120 + 0.992i)18-s + ⋯
L(s)  = 1  + (−0.399 − 0.916i)2-s + (−0.262 + 0.964i)3-s + (−0.681 + 0.732i)4-s + (−0.998 + 0.0483i)5-s + (0.989 − 0.144i)6-s + (0.168 + 0.985i)7-s + (0.943 + 0.331i)8-s + (−0.861 − 0.506i)9-s + (0.443 + 0.896i)10-s + (−0.527 − 0.849i)12-s + (0.443 − 0.896i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (−0.0724 − 0.997i)16-s + (−0.926 + 0.377i)17-s + (−0.120 + 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.986 + 0.163i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.986 + 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03297433880 + 0.4005784045i\)
\(L(\frac12)\) \(\approx\) \(0.03297433880 + 0.4005784045i\)
\(L(1)\) \(\approx\) \(0.5865502017 + 0.06681251138i\)
\(L(1)\) \(\approx\) \(0.5865502017 + 0.06681251138i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.399 - 0.916i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (0.168 + 0.985i)T \)
13 \( 1 + (0.443 - 0.896i)T \)
17 \( 1 + (-0.926 + 0.377i)T \)
19 \( 1 + (0.970 - 0.239i)T \)
23 \( 1 + (0.0241 + 0.999i)T \)
29 \( 1 + (0.861 - 0.506i)T \)
31 \( 1 + (-0.607 + 0.794i)T \)
37 \( 1 + (-0.906 - 0.421i)T \)
41 \( 1 + (0.0724 - 0.997i)T \)
43 \( 1 + (-0.779 + 0.626i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.644 + 0.764i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.779 + 0.626i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.354 + 0.935i)T \)
83 \( 1 + (-0.981 - 0.192i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.715 + 0.698i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.956713107478432747944073684828, −19.29719918869455372587311850002, −18.41758712051738772806685826143, −18.1078798030878557923273108845, −17.00538672055782919094612474841, −16.51951893390721209411935984273, −15.903208474969513929765874174136, −14.86242881739210959759564150127, −14.04192021475004311466088539722, −13.56997031545098137865833965542, −12.66943848426649443407685166147, −11.605016977222830216318411113675, −11.07448644519897396183242055858, −10.13528864563916663207574193107, −8.89680858041592284128868983179, −8.29752388942863473140784850883, −7.531040488966208843040573116759, −6.86791514777993688110301983724, −6.456693217550895249144449522, −5.10274754368139578728528578597, −4.453782397876922272204486718389, −3.4223754760673258431506495148, −1.83098840343122935842155404027, −0.849836503366151454528816108479, −0.14386565657946317948467825593, 0.97390279720965016900080019535, 2.4069227957030476376897942084, 3.325331221376048661685675914319, 3.82044372927010445283933937684, 4.93810991591973720709793895886, 5.460806197254426872673349069837, 6.88148778533495148444450557433, 8.14001646950006247630501761353, 8.54944313861399079351430845447, 9.35620418387233487375066030518, 10.18521705386720558962119931065, 11.090240944072220671647847275883, 11.45237554034640076078031226696, 12.19574158676059709682700752234, 12.95710952614160236443263531293, 14.09315169802184774640473776235, 15.05298316581758881030376086047, 15.82705457624762829987347743855, 16.07329556575246119626134912930, 17.501408576329007738699596534319, 17.787379611357342853416853834238, 18.712159392886192357312904144556, 19.70268873961553245141715654919, 19.97832987333570249345310466323, 20.95132902926268459382829191932

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