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Properties

Label 1-1049-1049.119-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.887 - 0.461i$
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  + (−0.985 − 0.167i)2-s + (0.976 − 0.214i)3-s + (0.944 + 0.329i)4-s + (−0.667 − 0.744i)5-s + (−0.998 + 0.0479i)6-s + (0.0599 + 0.998i)7-s + (−0.875 − 0.482i)8-s + (0.908 − 0.418i)9-s + (0.534 + 0.845i)10-s + (−0.631 − 0.775i)11-s + (0.992 + 0.119i)12-s + (0.363 + 0.931i)13-s + (0.107 − 0.994i)14-s + (−0.811 − 0.583i)15-s + (0.782 + 0.622i)16-s + (0.999 − 0.0239i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.167i)2-s + (0.976 − 0.214i)3-s + (0.944 + 0.329i)4-s + (−0.667 − 0.744i)5-s + (−0.998 + 0.0479i)6-s + (0.0599 + 0.998i)7-s + (−0.875 − 0.482i)8-s + (0.908 − 0.418i)9-s + (0.534 + 0.845i)10-s + (−0.631 − 0.775i)11-s + (0.992 + 0.119i)12-s + (0.363 + 0.931i)13-s + (0.107 − 0.994i)14-s + (−0.811 − 0.583i)15-s + (0.782 + 0.622i)16-s + (0.999 − 0.0239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.887 - 0.461i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.887 - 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240459382 - 0.3030505827i\)
\(L(\frac12)\) \(\approx\) \(1.240459382 - 0.3030505827i\)
\(L(1)\) \(\approx\) \(0.9282843038 - 0.1551826684i\)
\(L(1)\) \(\approx\) \(0.9282843038 - 0.1551826684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.985 - 0.167i)T \)
3 \( 1 + (0.976 - 0.214i)T \)
5 \( 1 + (-0.667 - 0.744i)T \)
7 \( 1 + (0.0599 + 0.998i)T \)
11 \( 1 + (-0.631 - 0.775i)T \)
13 \( 1 + (0.363 + 0.931i)T \)
17 \( 1 + (0.999 - 0.0239i)T \)
19 \( 1 + (0.178 - 0.983i)T \)
23 \( 1 + (0.295 + 0.955i)T \)
29 \( 1 + (-0.593 - 0.804i)T \)
31 \( 1 + (0.767 + 0.640i)T \)
37 \( 1 + (0.797 - 0.603i)T \)
41 \( 1 + (-0.178 + 0.983i)T \)
43 \( 1 + (-0.951 + 0.306i)T \)
47 \( 1 + (-0.908 + 0.418i)T \)
53 \( 1 + (0.958 - 0.283i)T \)
59 \( 1 + (-0.107 - 0.994i)T \)
61 \( 1 + (0.989 + 0.143i)T \)
67 \( 1 + (0.825 + 0.564i)T \)
71 \( 1 + (-0.998 - 0.0479i)T \)
73 \( 1 + (0.685 + 0.727i)T \)
79 \( 1 + (0.989 - 0.143i)T \)
83 \( 1 + (0.918 - 0.396i)T \)
89 \( 1 + (0.0599 - 0.998i)T \)
97 \( 1 + (0.407 - 0.913i)T \)
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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

−21.08954009089139258912829715960, −20.468224435942415846192933750139, −20.12404697404178706682767307153, −19.19027705949817425306069305297, −18.52338329584342451434741300493, −18.00438657606023809533861599308, −16.74418666662522211713351860450, −16.2062832312076071568693616041, −15.09772766142404118919464896198, −14.93707232941057674383322283892, −13.9853272394671478771787155067, −12.89944572690000724438678235069, −11.9403403749274197363151964652, −10.68292650468792661234382887792, −10.33292744999431858660230432128, −9.7573998326095916102144492314, −8.40008416564601486293392733842, −7.85544876711895948849339139451, −7.38451960453965128278576816519, −6.51475538134508235353397425120, −5.106693501258652099957029596765, −3.80000540606619877704291334561, −3.150616791006907399684407002427, −2.16473419382137604018924434794, −0.92006067841246902210490032236, 0.90591005834808682043769428065, 1.89044699356085480020763028424, 2.92048075032881578395112134959, 3.60899448423001945009219410351, 4.96658085180762081215938517769, 6.12369115067970926411127009228, 7.22763196755238950363039299716, 8.0726665857455070347432702593, 8.50317619426340024615005444916, 9.26560928687528548002951909114, 9.84301463894803971673825808441, 11.384007557082681080407861923090, 11.6586781111206103805383633905, 12.75041146854908684925940484352, 13.37308308969044175688629908426, 14.62190298559802664310951165432, 15.467299270902158300770914905243, 15.96536163861114915649753185342, 16.65119952420955240296607886923, 17.84578817596327898226967038949, 18.618855416226732426721967537970, 19.19686850670198488203815162233, 19.6158799797693115186101830697, 20.622795034373863121188184871446, 21.32199802763189953531444935739

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