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Properties

Label 1-1441-1441.1094-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.743 + 0.668i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (−0.861 − 0.506i)2-s + (−0.998 + 0.0483i)3-s + (0.485 + 0.873i)4-s + (0.836 + 0.548i)5-s + (0.885 + 0.464i)6-s + (−0.989 − 0.144i)7-s + (0.0241 − 0.999i)8-s + (0.995 − 0.0965i)9-s + (−0.443 − 0.896i)10-s + (−0.527 − 0.849i)12-s + (−0.168 − 0.985i)13-s + (0.779 + 0.626i)14-s + (−0.861 − 0.506i)15-s + (−0.527 + 0.849i)16-s + (−0.970 − 0.239i)17-s + (−0.906 − 0.421i)18-s + ⋯
L(s)  = 1  + (−0.861 − 0.506i)2-s + (−0.998 + 0.0483i)3-s + (0.485 + 0.873i)4-s + (0.836 + 0.548i)5-s + (0.885 + 0.464i)6-s + (−0.989 − 0.144i)7-s + (0.0241 − 0.999i)8-s + (0.995 − 0.0965i)9-s + (−0.443 − 0.896i)10-s + (−0.527 − 0.849i)12-s + (−0.168 − 0.985i)13-s + (0.779 + 0.626i)14-s + (−0.861 − 0.506i)15-s + (−0.527 + 0.849i)16-s + (−0.970 − 0.239i)17-s + (−0.906 − 0.421i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.743 + 0.668i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.743 + 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4559171592 + 0.1748576321i\)
\(L(\frac12)\) \(\approx\) \(0.4559171592 + 0.1748576321i\)
\(L(1)\) \(\approx\) \(0.5059615946 - 0.03623263292i\)
\(L(1)\) \(\approx\) \(0.5059615946 - 0.03623263292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.861 - 0.506i)T \)
3 \( 1 + (-0.998 + 0.0483i)T \)
5 \( 1 + (0.836 + 0.548i)T \)
7 \( 1 + (-0.989 - 0.144i)T \)
13 \( 1 + (-0.168 - 0.985i)T \)
17 \( 1 + (-0.970 - 0.239i)T \)
19 \( 1 + (-0.527 - 0.849i)T \)
23 \( 1 + (0.0241 + 0.999i)T \)
29 \( 1 + (0.215 + 0.976i)T \)
31 \( 1 + (0.0241 - 0.999i)T \)
37 \( 1 + (-0.681 + 0.732i)T \)
41 \( 1 + (-0.970 + 0.239i)T \)
43 \( 1 + (0.779 - 0.626i)T \)
47 \( 1 + (-0.861 - 0.506i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.926 - 0.377i)T \)
61 \( 1 + T \)
67 \( 1 + (0.779 + 0.626i)T \)
71 \( 1 + (0.0241 - 0.999i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.836 + 0.548i)T \)
83 \( 1 + (-0.681 - 0.732i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.989 - 0.144i)T \)
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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

−20.704255636449240716114072214310, −19.51724247997676172169701854407, −19.03953848519280513165882408856, −18.18566869677785573873840941169, −17.513603903299424032170185437450, −16.90220147626688067346981148677, −16.250165970088721921076268606571, −15.87831265811560608125965553788, −14.72156709723507541584755064831, −13.800030643307668559705934361439, −12.880124536282685864296305043765, −12.22152109925313648604450214646, −11.269038369737755297230899932350, −10.30984438617018142758118228401, −9.90649714628275005433129317692, −9.04878412896846096078590709391, −8.36747324837869947351532778963, −6.94937878894575713796467737931, −6.51669084075892065864724943815, −5.92848952275700843674023779663, −5.038507577519199061702761245141, −4.139362361205195220490351809426, −2.345259333768949695524379883961, −1.641031593736063500880123523686, −0.39143173528861812286646755443, 0.77802319385276250555315785740, 1.991059131771907407800106723663, 2.89447690030994990473896214435, 3.78473917601510798615985785905, 5.0806597698674169011276051891, 6.03331325275203437574116427704, 6.83791006369189617353693481959, 7.2118471857018854956574514563, 8.61032327609499145808207481517, 9.55870418763522756858473835127, 10.0359798993478807991400339434, 10.74275665684065516568636316933, 11.320704180400919355034334391130, 12.304417238523241961928114154385, 13.1671249220221864262052181069, 13.42734996056870590067187850209, 15.17549655551654772229870538324, 15.67099366504939154912185107381, 16.57101047096269263533075027621, 17.39008110326400105886765380269, 17.63604449363758037632515273274, 18.47619786515992894414114126846, 19.141952706734830360658530808159, 19.976510904207231463039681092130, 20.75835123212778295918476472144

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