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Properties

Label 1-1441-1441.291-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.366 + 0.930i$
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  + (0.262 − 0.964i)2-s + (−0.943 + 0.331i)3-s + (−0.861 − 0.506i)4-s + (−0.607 − 0.794i)5-s + (0.0724 + 0.997i)6-s + (−0.970 + 0.239i)7-s + (−0.715 + 0.698i)8-s + (0.779 − 0.626i)9-s + (−0.926 + 0.377i)10-s + (0.981 + 0.192i)12-s + (−0.970 + 0.239i)13-s + (−0.0241 + 0.999i)14-s + (0.836 + 0.548i)15-s + (0.485 + 0.873i)16-s + (−0.981 + 0.192i)17-s + (−0.399 − 0.916i)18-s + ⋯
L(s)  = 1  + (0.262 − 0.964i)2-s + (−0.943 + 0.331i)3-s + (−0.861 − 0.506i)4-s + (−0.607 − 0.794i)5-s + (0.0724 + 0.997i)6-s + (−0.970 + 0.239i)7-s + (−0.715 + 0.698i)8-s + (0.779 − 0.626i)9-s + (−0.926 + 0.377i)10-s + (0.981 + 0.192i)12-s + (−0.970 + 0.239i)13-s + (−0.0241 + 0.999i)14-s + (0.836 + 0.548i)15-s + (0.485 + 0.873i)16-s + (−0.981 + 0.192i)17-s + (−0.399 − 0.916i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.366 + 0.930i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.366 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2659338932 - 0.3906183204i\)
\(L(\frac12)\) \(\approx\) \(-0.2659338932 - 0.3906183204i\)
\(L(1)\) \(\approx\) \(0.4501251805 - 0.3693464117i\)
\(L(1)\) \(\approx\) \(0.4501251805 - 0.3693464117i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.262 - 0.964i)T \)
3 \( 1 + (-0.943 + 0.331i)T \)
5 \( 1 + (-0.607 - 0.794i)T \)
7 \( 1 + (-0.970 + 0.239i)T \)
13 \( 1 + (-0.970 + 0.239i)T \)
17 \( 1 + (-0.981 + 0.192i)T \)
19 \( 1 + (0.906 - 0.421i)T \)
23 \( 1 + (0.989 + 0.144i)T \)
29 \( 1 + (0.354 - 0.935i)T \)
31 \( 1 + (0.168 - 0.985i)T \)
37 \( 1 + (0.748 - 0.663i)T \)
41 \( 1 + (-0.681 - 0.732i)T \)
43 \( 1 + (-0.607 - 0.794i)T \)
47 \( 1 + (-0.779 + 0.626i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.981 - 0.192i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.607 - 0.794i)T \)
71 \( 1 + (0.443 + 0.896i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.0241 - 0.999i)T \)
83 \( 1 + (0.861 - 0.506i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.527 + 0.849i)T \)
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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

−21.63874967709416049305223644502, −20.0665859284902774297602050572, −19.35921636796419521848667557630, −18.51786409125337997864015404691, −18.0194210829595194228615694185, −17.177628453981987728635075516798, −16.42668334305980714870021962902, −15.89267331309956664374360523602, −15.16316458999749944155094845550, −14.33578634787284746607193505872, −13.40257788641345291955855196065, −12.74115573909080757339669051052, −12.0341069629976265265074829138, −11.19912159926162859181854378456, −10.189785147388980718556185602843, −9.57133979598740371630228540552, −8.28440456434953587886348732728, −7.421249948562058517401141992288, −6.71303886707121658087863990941, −6.52623002090586016849122318080, −5.24135494465131839509625374535, −4.65768613254480977730726856471, −3.51568805620072819277381031443, −2.75796585311583873187328150948, −0.88764519491858893613061645157, 0.19357745755989176352521853, 0.65331064563485724978979020106, 2.02647693555576768370747485997, 3.17663570451713042326139571968, 4.07263902213641296000960011608, 4.77549115224217973341307796036, 5.425478850503664763485517011516, 6.40060870658578378866870335961, 7.38682323138471216526967662682, 8.71979210750912134146238492260, 9.48058575155645526535583411107, 9.8995058294020827934262888809, 11.02295252064946039351065665587, 11.668167827630163342549778458724, 12.18720415345409706222626199356, 12.9982767885432439676131005392, 13.413542925895521951998817837668, 14.889469083517915235260124433220, 15.50740116183932308957279644063, 16.247176395721640852848507539779, 17.13771531166291338258960939240, 17.66411899754162998820120563836, 18.806714609556776148872223229282, 19.31770744727789604055322844414, 20.06140760462254948483472120790

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