↑

Properties

Label 1-1049-1049.11-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.971 - 0.238i$
Analytic cond. $4.87153$
Root an. cond. $4.87153$
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  + (−0.202 − 0.979i)2-s + (−0.804 + 0.593i)3-s + (−0.918 + 0.396i)4-s + (0.851 + 0.524i)5-s + (0.744 + 0.667i)6-s + (−0.260 − 0.965i)7-s + (0.574 + 0.818i)8-s + (0.295 − 0.955i)9-s + (0.340 − 0.940i)10-s + (−0.178 − 0.983i)11-s + (0.503 − 0.864i)12-s + (0.534 − 0.845i)13-s + (−0.892 + 0.450i)14-s + (−0.996 + 0.0838i)15-s + (0.685 − 0.727i)16-s + (0.913 − 0.407i)17-s + ⋯
L(s)  = 1  + (−0.202 − 0.979i)2-s + (−0.804 + 0.593i)3-s + (−0.918 + 0.396i)4-s + (0.851 + 0.524i)5-s + (0.744 + 0.667i)6-s + (−0.260 − 0.965i)7-s + (0.574 + 0.818i)8-s + (0.295 − 0.955i)9-s + (0.340 − 0.940i)10-s + (−0.178 − 0.983i)11-s + (0.503 − 0.864i)12-s + (0.534 − 0.845i)13-s + (−0.892 + 0.450i)14-s + (−0.996 + 0.0838i)15-s + (0.685 − 0.727i)16-s + (0.913 − 0.407i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.971 - 0.238i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.971 - 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08161138097 - 0.6751778843i\)
\(L(\frac12)\) \(\approx\) \(0.08161138097 - 0.6751778843i\)
\(L(1)\) \(\approx\) \(0.6086051952 - 0.3630988971i\)
\(L(1)\) \(\approx\) \(0.6086051952 - 0.3630988971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.202 - 0.979i)T \)
3 \( 1 + (-0.804 + 0.593i)T \)
5 \( 1 + (0.851 + 0.524i)T \)
7 \( 1 + (-0.260 - 0.965i)T \)
11 \( 1 + (-0.178 - 0.983i)T \)
13 \( 1 + (0.534 - 0.845i)T \)
17 \( 1 + (0.913 - 0.407i)T \)
19 \( 1 + (-0.702 - 0.711i)T \)
23 \( 1 + (-0.968 - 0.249i)T \)
29 \( 1 + (0.612 - 0.790i)T \)
31 \( 1 + (-0.922 + 0.385i)T \)
37 \( 1 + (-0.329 - 0.944i)T \)
41 \( 1 + (-0.711 + 0.702i)T \)
43 \( 1 + (-0.736 + 0.676i)T \)
47 \( 1 + (0.955 + 0.295i)T \)
53 \( 1 + (0.318 + 0.948i)T \)
59 \( 1 + (-0.450 + 0.892i)T \)
61 \( 1 + (0.811 - 0.583i)T \)
67 \( 1 + (-0.482 - 0.875i)T \)
71 \( 1 + (-0.744 + 0.667i)T \)
73 \( 1 + (0.131 - 0.991i)T \)
79 \( 1 + (-0.811 - 0.583i)T \)
83 \( 1 + (-0.658 + 0.752i)T \)
89 \( 1 + (0.260 - 0.965i)T \)
97 \( 1 + (0.272 - 0.962i)T \)
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

−21.96377955971727429776129134147, −21.505510524843169511998349830352, −20.32164412691464343344729591502, −19.01141861208506619087002139437, −18.59389576927068622275092866151, −17.89293549803840748308198707126, −17.186629742814514220420840778837, −16.511209834480836151519903590659, −15.91986513212529200065737063569, −14.86756516122762993945639245967, −14.04042532248396186610785718486, −13.18220402915346866382190318468, −12.488959280308734485210332825393, −11.9166140930335548629849758901, −10.35040985546729164681863349198, −9.889830557241874575625764897634, −8.816806291822286998731248703446, −8.18605199294779280663828088087, −7.02249079351257457274919553071, −6.3405789039606707005942837785, −5.61829927213167088067369160632, −5.102401811041740613485646119030, −3.9809253455663406943108619588, −2.0308729313691718609370930584, −1.45931777160619459629350856016, 0.36665482406426281538982304498, 1.36249751321976590707391127285, 2.8620710949126390837116840948, 3.503975790013361823095675140775, 4.45922622885903579664900744741, 5.53749307447804644623495061582, 6.1723439086794657144956038279, 7.36199591738372191955152255156, 8.51531627071288970637555117041, 9.52171541022429450636666977043, 10.255262360876277740934949424868, 10.668685842121861194482216017674, 11.27855193524381474598835457088, 12.34267408297889959486804846364, 13.24718996562018465046036346902, 13.825114084436589840603381859113, 14.6916228628348182467212217404, 15.97150666188441695039527056357, 16.72843742015625761807311019259, 17.37656419698207296209402280534, 18.085773125883269488796899629393, 18.685609915890158720117674377523, 19.75673749850591531332842407522, 20.546025049267152410739030610249, 21.318507000628634601590482115916

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