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Properties

Label 1-1441-1441.102-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.366 - 0.930i$
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  + (−0.354 − 0.935i)2-s + (−0.607 − 0.794i)3-s + (−0.748 + 0.663i)4-s + (0.0241 + 0.999i)5-s + (−0.527 + 0.849i)6-s + (−0.0724 + 0.997i)7-s + (0.885 + 0.464i)8-s + (−0.262 + 0.964i)9-s + (0.926 − 0.377i)10-s + (0.981 + 0.192i)12-s + (0.644 − 0.764i)13-s + (0.958 − 0.285i)14-s + (0.779 − 0.626i)15-s + (0.120 − 0.992i)16-s + (−0.906 − 0.421i)17-s + (0.995 − 0.0965i)18-s + ⋯
L(s)  = 1  + (−0.354 − 0.935i)2-s + (−0.607 − 0.794i)3-s + (−0.748 + 0.663i)4-s + (0.0241 + 0.999i)5-s + (−0.527 + 0.849i)6-s + (−0.0724 + 0.997i)7-s + (0.885 + 0.464i)8-s + (−0.262 + 0.964i)9-s + (0.926 − 0.377i)10-s + (0.981 + 0.192i)12-s + (0.644 − 0.764i)13-s + (0.958 − 0.285i)14-s + (0.779 − 0.626i)15-s + (0.120 − 0.992i)16-s + (−0.906 − 0.421i)17-s + (0.995 − 0.0965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.366 - 0.930i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.366 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6078224789 - 0.4139708667i\)
\(L(\frac12)\) \(\approx\) \(0.6078224789 - 0.4139708667i\)
\(L(1)\) \(\approx\) \(0.5974382245 - 0.2548342597i\)
\(L(1)\) \(\approx\) \(0.5974382245 - 0.2548342597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.354 - 0.935i)T \)
3 \( 1 + (-0.607 - 0.794i)T \)
5 \( 1 + (0.0241 + 0.999i)T \)
7 \( 1 + (-0.0724 + 0.997i)T \)
13 \( 1 + (0.644 - 0.764i)T \)
17 \( 1 + (-0.906 - 0.421i)T \)
19 \( 1 + (-0.681 - 0.732i)T \)
23 \( 1 + (-0.989 - 0.144i)T \)
29 \( 1 + (0.779 + 0.626i)T \)
31 \( 1 + (-0.443 - 0.896i)T \)
37 \( 1 + (0.399 + 0.916i)T \)
41 \( 1 + (0.485 - 0.873i)T \)
43 \( 1 + (-0.607 - 0.794i)T \)
47 \( 1 + (0.836 + 0.548i)T \)
53 \( 1 + T \)
59 \( 1 + (0.120 - 0.992i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.607 + 0.794i)T \)
71 \( 1 + (-0.168 + 0.985i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.607 - 0.794i)T \)
83 \( 1 + (0.995 + 0.0965i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.926 + 0.377i)T \)
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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

−21.05917880935430302932666845151, −19.89134592863032792405057507994, −19.61546076178507105216843834269, −18.142804034198916130969074719348, −17.695333128853276689133652294105, −16.74877350245001124946789428531, −16.50942779002818179456063101554, −15.89624563511404497258918644210, −15.05157521088385288744242923745, −14.12221468377993236761136493188, −13.44663497749118055978225722594, −12.57480876940495576406671243713, −11.52715283429586779751168931914, −10.57398917311495871795939624368, −10.036404250747335335055207964953, −9.112343288954214251443801303244, −8.56456135320420106292921098861, −7.64155746074880022156843373499, −6.46527292180411646897050052572, −6.054022926549515071591104328579, −4.99737877361005263617788214580, −4.12491980653727223990627232839, −3.986628096795528309632680006563, −1.7036220291099478187242225081, −0.6870663085233868440647855129, 0.5618833392069251913514913601, 2.06515028654248901464726895244, 2.41685167647990481456670329635, 3.39228667302372285767260534256, 4.611016150983958824425633566059, 5.66116682584228363437103061289, 6.41873572306654308652193527569, 7.30794907333255329070637937480, 8.241910418568906427431316517023, 8.88608992590181587399526876538, 10.0428823562237327815514057650, 10.77258127144645184777837032135, 11.36821369467641981668431737560, 11.98557910491483775657563153144, 12.83904124796942724281009418730, 13.44022855773236789832377575910, 14.253230883160350986364707869617, 15.36078852722770415264637243026, 16.09497773330076931484724534656, 17.3857335353222712690872477722, 17.76022987691641240687166917979, 18.5580231763215294816345715212, 18.75077388904874341906569132413, 19.71391352034921836784788707111, 20.364840390425544948057629384179

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