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Properties

Label 1-1441-1441.318-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.412 + 0.910i$
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  + (−0.0724 + 0.997i)2-s + (0.485 − 0.873i)3-s + (−0.989 − 0.144i)4-s + (0.981 + 0.192i)5-s + (0.836 + 0.548i)6-s + (−0.779 − 0.626i)7-s + (0.215 − 0.976i)8-s + (−0.527 − 0.849i)9-s + (−0.262 + 0.964i)10-s + (−0.607 + 0.794i)12-s + (0.262 + 0.964i)13-s + (0.681 − 0.732i)14-s + (0.644 − 0.764i)15-s + (0.958 + 0.285i)16-s + (0.0241 + 0.999i)17-s + (0.885 − 0.464i)18-s + ⋯
L(s)  = 1  + (−0.0724 + 0.997i)2-s + (0.485 − 0.873i)3-s + (−0.989 − 0.144i)4-s + (0.981 + 0.192i)5-s + (0.836 + 0.548i)6-s + (−0.779 − 0.626i)7-s + (0.215 − 0.976i)8-s + (−0.527 − 0.849i)9-s + (−0.262 + 0.964i)10-s + (−0.607 + 0.794i)12-s + (0.262 + 0.964i)13-s + (0.681 − 0.732i)14-s + (0.644 − 0.764i)15-s + (0.958 + 0.285i)16-s + (0.0241 + 0.999i)17-s + (0.885 − 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.412 + 0.910i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.412 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367969764 + 0.8821199147i\)
\(L(\frac12)\) \(\approx\) \(1.367969764 + 0.8821199147i\)
\(L(1)\) \(\approx\) \(1.123678748 + 0.3324536219i\)
\(L(1)\) \(\approx\) \(1.123678748 + 0.3324536219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.0724 + 0.997i)T \)
3 \( 1 + (0.485 - 0.873i)T \)
5 \( 1 + (0.981 + 0.192i)T \)
7 \( 1 + (-0.779 - 0.626i)T \)
13 \( 1 + (0.262 + 0.964i)T \)
17 \( 1 + (0.0241 + 0.999i)T \)
19 \( 1 + (0.568 + 0.822i)T \)
23 \( 1 + (-0.995 - 0.0965i)T \)
29 \( 1 + (-0.527 + 0.849i)T \)
31 \( 1 + (0.861 + 0.506i)T \)
37 \( 1 + (0.168 + 0.985i)T \)
41 \( 1 + (-0.958 + 0.285i)T \)
43 \( 1 + (0.906 - 0.421i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.943 + 0.331i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.906 + 0.421i)T \)
71 \( 1 + (0.748 - 0.663i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (0.715 - 0.698i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.998 + 0.0483i)T \)
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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

−20.4378716308138600279812115005, −20.20704527753244178051169430849, −19.22168865148193947843709287542, −18.44531817153841068807181648965, −17.70336142106012203611042615712, −16.95696330358134274641634570893, −15.94994089568810121440457041418, −15.37493911824666738970495854671, −14.23523896020113532048936117547, −13.63623371935463177034812562875, −13.077511234786278100713422824536, −12.1438946464409403474517761982, −11.250930729774593101153569621962, −10.34672836868438062782640665710, −9.73438879655714984144268178758, −9.28817843170818752201398597284, −8.592251680525172695858800187064, −7.57886651204092668972538274461, −5.94614148302377425953112248327, −5.438205577582778119143383729037, −4.53224037340992048242023300726, −3.504063927245758447951568246200, −2.66419144927045048131886074846, −2.2568745420768431534873067674, −0.67260543158870344534147202917, 1.10195858304168734275734642791, 1.9318830964975346970456760686, 3.30063639005227276624090943802, 4.0096697625305128649242315727, 5.37605725470036079411197118522, 6.35047677884095661262272898755, 6.52506480210010497831428521666, 7.44612813800694157108099737196, 8.31483577312132610494890676861, 9.09921309472197995876984507159, 9.833689616566692133977842048082, 10.50201526553549210428455243002, 12.023854604203456107001134399450, 12.823857696227007287539681042, 13.50864564386699114991024357850, 14.042420783139406975005191431631, 14.512238728392877188459206404739, 15.567310161707705670959377021688, 16.563525307785247734346378414130, 17.03508500779720344247702223308, 17.824062732303195189222845132666, 18.59363152656564889309970476056, 19.02351442413996067714835965505, 19.954655488103015034215481425395, 20.81174856752637312543434009112

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