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Properties

Label 1-1441-1441.1069-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.0949 + 0.995i$
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  + (−0.779 + 0.626i)2-s + (0.568 − 0.822i)3-s + (0.215 − 0.976i)4-s + (0.568 − 0.822i)5-s + (0.0724 + 0.997i)6-s + (0.0724 − 0.997i)7-s + (0.443 + 0.896i)8-s + (−0.354 − 0.935i)9-s + (0.0724 + 0.997i)10-s + (−0.681 − 0.732i)12-s + (−0.644 + 0.764i)13-s + (0.568 + 0.822i)14-s + (−0.354 − 0.935i)15-s + (−0.906 − 0.421i)16-s + (−0.981 + 0.192i)17-s + (0.861 + 0.506i)18-s + ⋯
L(s)  = 1  + (−0.779 + 0.626i)2-s + (0.568 − 0.822i)3-s + (0.215 − 0.976i)4-s + (0.568 − 0.822i)5-s + (0.0724 + 0.997i)6-s + (0.0724 − 0.997i)7-s + (0.443 + 0.896i)8-s + (−0.354 − 0.935i)9-s + (0.0724 + 0.997i)10-s + (−0.681 − 0.732i)12-s + (−0.644 + 0.764i)13-s + (0.568 + 0.822i)14-s + (−0.354 − 0.935i)15-s + (−0.906 − 0.421i)16-s + (−0.981 + 0.192i)17-s + (0.861 + 0.506i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0949 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0949 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.0949 + 0.995i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.0949 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07807030516 - 0.08587418843i\)
\(L(\frac12)\) \(\approx\) \(-0.07807030516 - 0.08587418843i\)
\(L(1)\) \(\approx\) \(0.7148820629 - 0.2866846164i\)
\(L(1)\) \(\approx\) \(0.7148820629 - 0.2866846164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.779 + 0.626i)T \)
3 \( 1 + (0.568 - 0.822i)T \)
5 \( 1 + (0.568 - 0.822i)T \)
7 \( 1 + (0.0724 - 0.997i)T \)
13 \( 1 + (-0.644 + 0.764i)T \)
17 \( 1 + (-0.981 + 0.192i)T \)
19 \( 1 + (-0.981 - 0.192i)T \)
23 \( 1 + (-0.168 - 0.985i)T \)
29 \( 1 + (-0.779 - 0.626i)T \)
31 \( 1 + (0.885 + 0.464i)T \)
37 \( 1 + (0.399 + 0.916i)T \)
41 \( 1 + (0.681 + 0.732i)T \)
43 \( 1 + (-0.0241 - 0.999i)T \)
47 \( 1 + (-0.998 + 0.0483i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.485 + 0.873i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.0241 - 0.999i)T \)
71 \( 1 + (-0.989 + 0.144i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.943 - 0.331i)T \)
83 \( 1 + (0.748 + 0.663i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.970 + 0.239i)T \)
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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

−21.20745005286488518663401649285, −20.33805577285133877634997978700, −19.37332457849755484455720614174, −19.15128692760087165757784541139, −17.907810760090058074409503977955, −17.71014848705916679688871738740, −16.64858170287275109821968531466, −15.73092466220827181897543088825, −15.10807582581216615478824010533, −14.47805363712786365872529314360, −13.33407923863990820035243547469, −12.735791804312925079338165416780, −11.48899388577264948072634263142, −11.01439854246268490411919748402, −10.14036640114957878669378177273, −9.57948440402342127198583888933, −8.91536194839133171150582689169, −8.11049116058796992070317434039, −7.27924124288433569925030653614, −6.15219903079068372954952315786, −5.18822436380586279851807070275, −4.05646626425125812101675143256, −3.10168313783637120651635045042, −2.430688184142193706684118625310, −1.89075762550674800316285773796, 0.02831692593506217690379861393, 0.87281867017476878045586061501, 1.81275782708689450510769964297, 2.45081208043765864976455974043, 4.211607530840454265851458986154, 4.84800249610910508602160931837, 6.32803871664543127104169080333, 6.53377303182101635828195137022, 7.559765694061724780405750506302, 8.284508710298367315518976999429, 8.94622407450702236517625917030, 9.64918600062467105913527279562, 10.477611256438663503227479240181, 11.47711461156851772379247591166, 12.50106836462158855703979941520, 13.38833069356638108244621496967, 13.85531517651621156319342793990, 14.65303321260820923643921369791, 15.41439336688094316050894986286, 16.67569988846544926488626634057, 16.925444127361035277855993930898, 17.66897386445449556171585202652, 18.34366247086943203628187126069, 19.39385633649768846639664520152, 19.69604882386421121910542397612

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