↑

Properties

Label 1-1441-1441.46-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.991 - 0.127i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.215 + 0.976i)2-s + (0.836 + 0.548i)3-s + (−0.906 − 0.421i)4-s + (0.779 − 0.626i)5-s + (−0.715 + 0.698i)6-s + (−0.885 + 0.464i)7-s + (0.607 − 0.794i)8-s + (0.399 + 0.916i)9-s + (0.443 + 0.896i)10-s + (−0.527 − 0.849i)12-s + (−0.885 + 0.464i)13-s + (−0.262 − 0.964i)14-s + (0.995 − 0.0965i)15-s + (0.644 + 0.764i)16-s + (0.527 − 0.849i)17-s + (−0.981 + 0.192i)18-s + ⋯
L(s)  = 1  + (−0.215 + 0.976i)2-s + (0.836 + 0.548i)3-s + (−0.906 − 0.421i)4-s + (0.779 − 0.626i)5-s + (−0.715 + 0.698i)6-s + (−0.885 + 0.464i)7-s + (0.607 − 0.794i)8-s + (0.399 + 0.916i)9-s + (0.443 + 0.896i)10-s + (−0.527 − 0.849i)12-s + (−0.885 + 0.464i)13-s + (−0.262 − 0.964i)14-s + (0.995 − 0.0965i)15-s + (0.644 + 0.764i)16-s + (0.527 − 0.849i)17-s + (−0.981 + 0.192i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.991 - 0.127i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1032599494 + 1.612366428i\)
\(L(\frac12)\) \(\approx\) \(-0.1032599494 + 1.612366428i\)
\(L(1)\) \(\approx\) \(0.8852572470 + 0.6875229788i\)
\(L(1)\) \(\approx\) \(0.8852572470 + 0.6875229788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.215 + 0.976i)T \)
3 \( 1 + (0.836 + 0.548i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (-0.885 + 0.464i)T \)
13 \( 1 + (-0.885 + 0.464i)T \)
17 \( 1 + (0.527 - 0.849i)T \)
19 \( 1 + (0.0724 - 0.997i)T \)
23 \( 1 + (0.0241 + 0.999i)T \)
29 \( 1 + (0.748 + 0.663i)T \)
31 \( 1 + (0.958 - 0.285i)T \)
37 \( 1 + (0.120 - 0.992i)T \)
41 \( 1 + (-0.926 - 0.377i)T \)
43 \( 1 + (-0.779 + 0.626i)T \)
47 \( 1 + (0.399 + 0.916i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.779 + 0.626i)T \)
71 \( 1 + (-0.943 - 0.331i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.262 - 0.964i)T \)
83 \( 1 + (0.906 - 0.421i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.168 - 0.985i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.148393375835251687892118360719, −19.28104843007764410668357668923, −18.87746809980104965755013985776, −18.21721722530940957960631223244, −17.242650884613156522397817873056, −16.82719903516916648815618320157, −15.331121476383318150690183853537, −14.50106215124290079983395331482, −13.89583101078053372656359395091, −13.251793029480791855232115469734, −12.51984695851899996792797668567, −11.97742797570817886853899753613, −10.53024131499413747598082193412, −10.01626580679503376509546782729, −9.67790436212485303844206495419, −8.46966523714336373002817838509, −7.88024811339136499102839427025, −6.81134600275809862107333134596, −6.12485273974363889297304330593, −4.792032792216430877161123384923, −3.57061456723909880565803723541, −3.09382827439397915960745972028, −2.268773142855764991043184299814, −1.42564880317929917640698734833, −0.30468635240082970098778763403, 1.07805858673966807134398559637, 2.37231676370329286728134882147, 3.23396269574386710240993771824, 4.532001023031778233021938568775, 5.04522696679346607680712724930, 5.91447359396565492366521571561, 6.91739290298513337346915195340, 7.64961598745083694323088226870, 8.75399062992909502284695503326, 9.240239716098664108635786312358, 9.711353080047934887845628362828, 10.38012627330894163279693066143, 11.94145632810298887962958040673, 12.916487466808647044259151986208, 13.58040792245565623207087908864, 14.11748921263436856404683465288, 14.96725450352375674882877601709, 15.80316404752674743874411591004, 16.211430817390896471511519180043, 17.00128064496190207926214585412, 17.74480481830265354136224757998, 18.698510279072874834293556217699, 19.44164868195952861282026716095, 19.966550322750352569250164658692, 21.06952481974410432000468450715

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