Properties

Label 1-1441-1441.1091-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.409 - 0.912i$
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  + (0.943 + 0.331i)2-s + (−0.989 + 0.144i)3-s + (0.779 + 0.626i)4-s + (−0.168 + 0.985i)5-s + (−0.981 − 0.192i)6-s + (−0.120 + 0.992i)7-s + (0.527 + 0.849i)8-s + (0.958 − 0.285i)9-s + (−0.485 + 0.873i)10-s + (−0.861 − 0.506i)12-s + (−0.120 + 0.992i)13-s + (−0.443 + 0.896i)14-s + (0.0241 − 0.999i)15-s + (0.215 + 0.976i)16-s + (0.861 − 0.506i)17-s + (0.998 + 0.0483i)18-s + ⋯
L(s)  = 1  + (0.943 + 0.331i)2-s + (−0.989 + 0.144i)3-s + (0.779 + 0.626i)4-s + (−0.168 + 0.985i)5-s + (−0.981 − 0.192i)6-s + (−0.120 + 0.992i)7-s + (0.527 + 0.849i)8-s + (0.958 − 0.285i)9-s + (−0.485 + 0.873i)10-s + (−0.861 − 0.506i)12-s + (−0.120 + 0.992i)13-s + (−0.443 + 0.896i)14-s + (0.0241 − 0.999i)15-s + (0.215 + 0.976i)16-s + (0.861 − 0.506i)17-s + (0.998 + 0.0483i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.409 - 0.912i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.409 - 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8368074929 + 1.292077249i\)
\(L(\frac12)\) \(\approx\) \(-0.8368074929 + 1.292077249i\)
\(L(1)\) \(\approx\) \(0.9148987119 + 0.8482114001i\)
\(L(1)\) \(\approx\) \(0.9148987119 + 0.8482114001i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.943 + 0.331i)T \)
3 \( 1 + (-0.989 + 0.144i)T \)
5 \( 1 + (-0.168 + 0.985i)T \)
7 \( 1 + (-0.120 + 0.992i)T \)
13 \( 1 + (-0.120 + 0.992i)T \)
17 \( 1 + (0.861 - 0.506i)T \)
19 \( 1 + (-0.399 - 0.916i)T \)
23 \( 1 + (0.926 - 0.377i)T \)
29 \( 1 + (-0.568 + 0.822i)T \)
31 \( 1 + (-0.0724 + 0.997i)T \)
37 \( 1 + (-0.354 + 0.935i)T \)
41 \( 1 + (-0.995 + 0.0965i)T \)
43 \( 1 + (0.168 - 0.985i)T \)
47 \( 1 + (0.958 - 0.285i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.861 + 0.506i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.168 - 0.985i)T \)
71 \( 1 + (0.644 - 0.764i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.443 + 0.896i)T \)
83 \( 1 + (-0.779 + 0.626i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.906 - 0.421i)T \)
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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

−20.33597731123792752324253744111, −19.32887860727641482309669383927, −18.79979767090719590665571637996, −17.35907034330317055492501719136, −17.002892998492244616221352907165, −16.28615986800361544980974492889, −15.53708104620166921040126610004, −14.690421944311991571123175024600, −13.61522166248981179332902229266, −12.884788478341621233273330593285, −12.60836507542382013980244930474, −11.70481661410458190554719489539, −10.93229535511681857046376199097, −10.23444206842070242687802869268, −9.56204919766285265291978483468, −7.89055492874851476800299822376, −7.46890601035457488811042632107, −6.25290022583322707077706734336, −5.64412855221674780841584218184, −4.90981625996402737480360130262, −4.09571564931098267518954957565, −3.42659658629238730374215935771, −1.814862549421854053391463455086, −1.03524124446797987657460595054, −0.24821382491221509673331461391, 1.606874345805855225107094058804, 2.70216486648747203251519748369, 3.44378782088381744861149207356, 4.56577551264726888437498610055, 5.245830186976037668312382236378, 6.02622272290254320346714761697, 6.9609234439227256028229509925, 7.07935031416803148293176849277, 8.54510032824343722473169474402, 9.55565925696833050741154293797, 10.71604982971858764309578237465, 11.17095184447261537657728382283, 12.098568418960130809470789696948, 12.32171682384408936421988273812, 13.520971847322749503653076463375, 14.28718250011245291919752065871, 15.27508269443842990100562551773, 15.425972671600525883468086123593, 16.51655376350742254872842368100, 16.99662170570336416248171962350, 18.080135402218762532723554235428, 18.68583603084587906343558597960, 19.38946289654721885096771037604, 20.69080010217484043622446341663, 21.483989855380089623214594798454

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