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Properties

Label 1-1049-1049.7-r1-0-0
Degree $1$
Conductor $1049$
Sign $0.632 + 0.774i$
Analytic cond. $112.730$
Root an. cond. $112.730$
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  + (0.992 + 0.119i)2-s + (0.0419 + 0.999i)3-s + (0.971 + 0.237i)4-s + (0.564 − 0.825i)5-s + (−0.0778 + 0.996i)6-s + (0.957 + 0.289i)7-s + (0.935 + 0.352i)8-s + (−0.996 + 0.0838i)9-s + (0.658 − 0.752i)10-s + (−0.260 − 0.965i)11-s + (−0.196 + 0.980i)12-s + (0.918 + 0.396i)13-s + (0.915 + 0.401i)14-s + (0.848 + 0.529i)15-s + (0.887 + 0.461i)16-s + (0.346 + 0.938i)17-s + ⋯
L(s)  = 1  + (0.992 + 0.119i)2-s + (0.0419 + 0.999i)3-s + (0.971 + 0.237i)4-s + (0.564 − 0.825i)5-s + (−0.0778 + 0.996i)6-s + (0.957 + 0.289i)7-s + (0.935 + 0.352i)8-s + (−0.996 + 0.0838i)9-s + (0.658 − 0.752i)10-s + (−0.260 − 0.965i)11-s + (−0.196 + 0.980i)12-s + (0.918 + 0.396i)13-s + (0.915 + 0.401i)14-s + (0.848 + 0.529i)15-s + (0.887 + 0.461i)16-s + (0.346 + 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.632 + 0.774i$
Analytic conductor: \(112.730\)
Root analytic conductor: \(112.730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (1:\ ),\ 0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.832897487 + 2.765954119i\)
\(L(\frac12)\) \(\approx\) \(5.832897487 + 2.765954119i\)
\(L(1)\) \(\approx\) \(2.555367193 + 0.7812491570i\)
\(L(1)\) \(\approx\) \(2.555367193 + 0.7812491570i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.992 + 0.119i)T \)
3 \( 1 + (0.0419 + 0.999i)T \)
5 \( 1 + (0.564 - 0.825i)T \)
7 \( 1 + (0.957 + 0.289i)T \)
11 \( 1 + (-0.260 - 0.965i)T \)
13 \( 1 + (0.918 + 0.396i)T \)
17 \( 1 + (0.346 + 0.938i)T \)
19 \( 1 + (0.995 - 0.0957i)T \)
23 \( 1 + (0.101 + 0.994i)T \)
29 \( 1 + (0.429 - 0.903i)T \)
31 \( 1 + (-0.672 - 0.740i)T \)
37 \( 1 + (-0.992 + 0.125i)T \)
41 \( 1 + (0.771 + 0.636i)T \)
43 \( 1 + (-0.782 + 0.622i)T \)
47 \( 1 + (0.645 - 0.763i)T \)
53 \( 1 + (0.450 - 0.892i)T \)
59 \( 1 + (-0.931 - 0.363i)T \)
61 \( 1 + (-0.851 - 0.524i)T \)
67 \( 1 + (0.549 + 0.835i)T \)
71 \( 1 + (0.996 - 0.0778i)T \)
73 \( 1 + (0.513 + 0.857i)T \)
79 \( 1 + (0.524 + 0.851i)T \)
83 \( 1 + (0.963 + 0.266i)T \)
89 \( 1 + (-0.289 - 0.957i)T \)
97 \( 1 + (0.295 - 0.955i)T \)
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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

−21.13854060969635372177467474657, −20.47819426211497682095279536838, −20.01584611643112748960926716901, −18.702283650672910677669192857659, −18.18617692867321450188891057258, −17.59589326645721414674099537897, −16.503277987924532824312411362610, −15.41870526920373539215343285142, −14.602624678286116837527610148382, −13.936530240460733075191393748677, −13.636017404975140247926631188705, −12.45453331643391844036486550590, −11.97911230712413269651430273590, −10.8132846254677056572289883638, −10.59497188264071075672654244163, −9.1313512687477700170932588085, −7.788020523251011043504722378248, −7.25380507373444932147879458113, −6.5709444946380670240052076421, −5.5358295628505729201571024407, −4.948901869438476480932245099283, −3.52434065531740598928240852431, −2.70089388509973469064801620742, −1.83692514759202100010255985656, −1.04912333398598845212840538691, 1.13726829205952259921238955591, 2.142605234392338625883619114365, 3.37907487740963636515070764040, 4.07640618034237698042493765377, 5.084037752019917369925363799033, 5.5665458036217363726060862359, 6.22007579753080033278361831571, 7.91346407149787944590311520242, 8.42809098800829819830743839452, 9.39224218454835806586130006560, 10.42860242014041156207676265115, 11.33246802688264676375671186190, 11.712749654213232230570495449560, 12.93282472473871040432430969557, 13.82910905801164272238592708514, 14.14056673693304173937492505173, 15.29197438934427767796499023366, 15.77407264992821282776102326923, 16.65187466331738472046387059823, 17.12463172943286429386584172396, 18.180920887952471134766574999674, 19.470555003582073684889213777749, 20.34046574464755508390088679858, 21.0045743111886057558543716492, 21.468069841667263666573791710746

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