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Properties

Label 1-1441-1441.171-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.861 - 0.507i$
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  + (−0.168 + 0.985i)2-s + (−0.527 + 0.849i)3-s + (−0.943 − 0.331i)4-s + (0.926 − 0.377i)5-s + (−0.748 − 0.663i)6-s + (−0.995 + 0.0965i)7-s + (0.485 − 0.873i)8-s + (−0.443 − 0.896i)9-s + (0.215 + 0.976i)10-s + (0.779 − 0.626i)12-s + (−0.399 − 0.916i)13-s + (0.0724 − 0.997i)14-s + (−0.168 + 0.985i)15-s + (0.779 + 0.626i)16-s + (−0.354 + 0.935i)17-s + (0.958 − 0.285i)18-s + ⋯
L(s)  = 1  + (−0.168 + 0.985i)2-s + (−0.527 + 0.849i)3-s + (−0.943 − 0.331i)4-s + (0.926 − 0.377i)5-s + (−0.748 − 0.663i)6-s + (−0.995 + 0.0965i)7-s + (0.485 − 0.873i)8-s + (−0.443 − 0.896i)9-s + (0.215 + 0.976i)10-s + (0.779 − 0.626i)12-s + (−0.399 − 0.916i)13-s + (0.0724 − 0.997i)14-s + (−0.168 + 0.985i)15-s + (0.779 + 0.626i)16-s + (−0.354 + 0.935i)17-s + (0.958 − 0.285i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.861 - 0.507i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.861 - 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1045958039 + 0.3836967779i\)
\(L(\frac12)\) \(\approx\) \(-0.1045958039 + 0.3836967779i\)
\(L(1)\) \(\approx\) \(0.5205768905 + 0.3930054609i\)
\(L(1)\) \(\approx\) \(0.5205768905 + 0.3930054609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.168 + 0.985i)T \)
3 \( 1 + (-0.527 + 0.849i)T \)
5 \( 1 + (0.926 - 0.377i)T \)
7 \( 1 + (-0.995 + 0.0965i)T \)
13 \( 1 + (-0.399 - 0.916i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (0.779 - 0.626i)T \)
23 \( 1 + (-0.485 - 0.873i)T \)
29 \( 1 + (-0.989 - 0.144i)T \)
31 \( 1 + (-0.485 + 0.873i)T \)
37 \( 1 + (-0.0241 + 0.999i)T \)
41 \( 1 + (0.354 + 0.935i)T \)
43 \( 1 + (0.0724 + 0.997i)T \)
47 \( 1 + (0.168 - 0.985i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.262 - 0.964i)T \)
61 \( 1 - T \)
67 \( 1 + (0.0724 - 0.997i)T \)
71 \( 1 + (-0.485 + 0.873i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.926 - 0.377i)T \)
83 \( 1 + (0.0241 + 0.999i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.995 + 0.0965i)T \)
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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

−20.20762063346134070520756902527, −19.34065624181112622268104318359, −18.80813968947095551134022443614, −18.19222821739090077528060584713, −17.53569244148558674487652955316, −16.75596527563317049282394049942, −16.14656612490875432284907176617, −14.553083964920320279083471580404, −13.70684606991874175287763240247, −13.477758986032748926332745310490, −12.54116855309365657476467266469, −11.89033026595344920344030105499, −11.1322716484293952086126393697, −10.31895327783890600477615110744, −9.45026548074528489789765526137, −9.09691407432376576925313055528, −7.52062290078484465795317177968, −7.10099228528627101392407915628, −5.89018570438708005545864607388, −5.42601593207693288168236018777, −4.11709883696362540131086258568, −3.051401026988665321533468992655, −2.211185388638213077039525135561, −1.55144140937305897669991706685, −0.1921220967926416582125656010, 1.090199484579979048464281210801, 2.77527475953914055907092358075, 3.79642356187036121175233377047, 4.81887248197678408514435333503, 5.448371727818002098716063497248, 6.18500645541658367060898943480, 6.70495059571589194810298974516, 7.992774485264866773515272190042, 8.97230106128261855640325496077, 9.49871292477878083301839162077, 10.16468731467365788022368946966, 10.75727602620886914136236956764, 12.25704356884321204403391084235, 12.91084600050493266346890196225, 13.59935866231795931745419469820, 14.64108343756480377584862061777, 15.24258945257788272229064794152, 16.02618316972584287854062029854, 16.62512856627217099635702967569, 17.199398094416135590523908444255, 17.89873674722889060571448246234, 18.52979003447235701510034094712, 19.82530199235725887240882052778, 20.291482575339411523405853102592, 21.61833446654614738167678161252

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