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Properties

Label 1-1049-1049.201-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.0533 - 0.998i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.719 − 0.694i)2-s + (0.935 − 0.352i)3-s + (0.0359 − 0.999i)4-s + (−0.767 + 0.640i)5-s + (0.429 − 0.903i)6-s + (−0.811 + 0.583i)7-s + (−0.667 − 0.744i)8-s + (0.752 − 0.658i)9-s + (−0.107 + 0.994i)10-s + (0.908 + 0.418i)11-s + (−0.318 − 0.948i)12-s + (0.995 − 0.0957i)13-s + (−0.178 + 0.983i)14-s + (−0.493 + 0.869i)15-s + (−0.997 − 0.0718i)16-s + (−0.534 − 0.845i)17-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)2-s + (0.935 − 0.352i)3-s + (0.0359 − 0.999i)4-s + (−0.767 + 0.640i)5-s + (0.429 − 0.903i)6-s + (−0.811 + 0.583i)7-s + (−0.667 − 0.744i)8-s + (0.752 − 0.658i)9-s + (−0.107 + 0.994i)10-s + (0.908 + 0.418i)11-s + (−0.318 − 0.948i)12-s + (0.995 − 0.0957i)13-s + (−0.178 + 0.983i)14-s + (−0.493 + 0.869i)15-s + (−0.997 − 0.0718i)16-s + (−0.534 − 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0533 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0533 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.0533 - 0.998i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.0533 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.941207450 - 1.840333238i\)
\(L(\frac12)\) \(\approx\) \(1.941207450 - 1.840333238i\)
\(L(1)\) \(\approx\) \(1.606913240 - 0.8565257322i\)
\(L(1)\) \(\approx\) \(1.606913240 - 0.8565257322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.719 - 0.694i)T \)
3 \( 1 + (0.935 - 0.352i)T \)
5 \( 1 + (-0.767 + 0.640i)T \)
7 \( 1 + (-0.811 + 0.583i)T \)
11 \( 1 + (0.908 + 0.418i)T \)
13 \( 1 + (0.995 - 0.0957i)T \)
17 \( 1 + (-0.534 - 0.845i)T \)
19 \( 1 + (-0.295 + 0.955i)T \)
23 \( 1 + (0.999 + 0.0239i)T \)
29 \( 1 + (-0.0119 - 0.999i)T \)
31 \( 1 + (0.593 - 0.804i)T \)
37 \( 1 + (0.472 - 0.881i)T \)
41 \( 1 + (0.295 - 0.955i)T \)
43 \( 1 + (0.864 + 0.503i)T \)
47 \( 1 + (-0.752 + 0.658i)T \)
53 \( 1 + (0.887 - 0.461i)T \)
59 \( 1 + (0.178 + 0.983i)T \)
61 \( 1 + (0.971 + 0.237i)T \)
67 \( 1 + (-0.998 + 0.0479i)T \)
71 \( 1 + (0.429 + 0.903i)T \)
73 \( 1 + (-0.951 - 0.306i)T \)
79 \( 1 + (0.971 - 0.237i)T \)
83 \( 1 + (0.155 + 0.987i)T \)
89 \( 1 + (-0.811 - 0.583i)T \)
97 \( 1 + (-0.340 - 0.940i)T \)
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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

−21.74444675256841148184578528476, −21.01344408840275153931823209018, −20.11916129877621988886441488494, −19.67032010339418126464379780969, −18.891096328110961018365087285461, −17.494329248564472816879166186613, −16.589900069146516323773688896726, −16.176313620620429554756581748815, −15.39143163190746143842871541284, −14.79741610456485442795560502614, −13.770293427309794149702882134382, −13.19523140705880130895616300193, −12.65675559772327072313293971255, −11.4622155736020643814986095251, −10.658666490706070758852545581052, −9.15231064956694138246041898751, −8.79312774313683383802594126952, −8.03638973264179843314398936058, −6.95028206746843036935082847408, −6.441971570543861358323835644767, −5.00171083910174916327474573699, −4.183413724335721348236670790917, −3.63209011558315803139201211881, −2.91928163774216527736551249385, −1.232901335243505743446585497846, 0.91745447358002817279621988550, 2.24889046410359480507003275383, 2.88288927624941982490785203916, 3.8119821548422915909058357041, 4.24175894263457287592897573016, 5.91547099084737416577355329599, 6.58432480073086937772236585373, 7.38257547337009175193107682315, 8.62603651346279273807314725969, 9.34268159256334760929252960788, 10.118979600943972107877637411231, 11.24097685407607164828195638373, 11.900823200409074127117377221198, 12.687598916034959043805748775745, 13.38432855415122181529767902031, 14.23461061004684545010033885524, 14.934733833932044533432983121342, 15.50570243492297016859992842575, 16.22634621785585071964703676880, 17.93575591364374312472171411058, 18.618157150127144714722966013768, 19.374969049394519362915181077193, 19.52162262187294794213591633565, 20.68728297851876986215320258631, 21.06004857630395118318484641471

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