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Properties

Label 1-1049-1049.1005-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.168 + 0.985i$
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  + (−0.767 − 0.640i)2-s + (0.974 + 0.225i)3-s + (0.178 + 0.983i)4-s + (−0.944 − 0.329i)5-s + (−0.603 − 0.797i)6-s + (0.0239 + 0.999i)7-s + (0.493 − 0.869i)8-s + (0.897 + 0.440i)9-s + (0.513 + 0.857i)10-s + (0.554 + 0.832i)11-s + (−0.0479 + 0.998i)12-s + (0.887 + 0.461i)13-s + (0.622 − 0.782i)14-s + (−0.845 − 0.534i)15-s + (−0.935 + 0.352i)16-s + (−0.948 + 0.318i)17-s + ⋯
L(s)  = 1  + (−0.767 − 0.640i)2-s + (0.974 + 0.225i)3-s + (0.178 + 0.983i)4-s + (−0.944 − 0.329i)5-s + (−0.603 − 0.797i)6-s + (0.0239 + 0.999i)7-s + (0.493 − 0.869i)8-s + (0.897 + 0.440i)9-s + (0.513 + 0.857i)10-s + (0.554 + 0.832i)11-s + (−0.0479 + 0.998i)12-s + (0.887 + 0.461i)13-s + (0.622 − 0.782i)14-s + (−0.845 − 0.534i)15-s + (−0.935 + 0.352i)16-s + (−0.948 + 0.318i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.168 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5930364168 + 0.7032471839i\)
\(L(\frac12)\) \(\approx\) \(0.5930364168 + 0.7032471839i\)
\(L(1)\) \(\approx\) \(0.8285177523 + 0.1205235212i\)
\(L(1)\) \(\approx\) \(0.8285177523 + 0.1205235212i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.767 - 0.640i)T \)
3 \( 1 + (0.974 + 0.225i)T \)
5 \( 1 + (-0.944 - 0.329i)T \)
7 \( 1 + (0.0239 + 0.999i)T \)
11 \( 1 + (0.554 + 0.832i)T \)
13 \( 1 + (0.887 + 0.461i)T \)
17 \( 1 + (-0.948 + 0.318i)T \)
19 \( 1 + (-0.997 - 0.0718i)T \)
23 \( 1 + (0.119 + 0.992i)T \)
29 \( 1 + (-0.0599 + 0.998i)T \)
31 \( 1 + (-0.999 + 0.0359i)T \)
37 \( 1 + (-0.775 - 0.631i)T \)
41 \( 1 + (0.0718 - 0.997i)T \)
43 \( 1 + (-0.875 - 0.482i)T \)
47 \( 1 + (0.440 - 0.897i)T \)
53 \( 1 + (-0.736 + 0.676i)T \)
59 \( 1 + (-0.782 + 0.622i)T \)
61 \( 1 + (0.363 - 0.931i)T \)
67 \( 1 + (0.237 - 0.971i)T \)
71 \( 1 + (0.603 - 0.797i)T \)
73 \( 1 + (-0.0119 + 0.999i)T \)
79 \( 1 + (-0.363 - 0.931i)T \)
83 \( 1 + (0.711 + 0.702i)T \)
89 \( 1 + (-0.0239 + 0.999i)T \)
97 \( 1 + (-0.985 - 0.167i)T \)
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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

−20.87358845822146464815846686375, −20.230298356316406355312236246873, −19.65537060263673166532968608324, −18.982694924240434199768304274524, −18.41046481528382668789569935682, −17.44496603586359567974568011253, −16.48221535169068179751980900789, −15.87029183261414929409354371583, −15.055519321831073389062699683680, −14.42990167949220131416765568339, −13.66150166191566003573821101667, −12.885224033346906159557355566573, −11.38394401093795990504827920979, −10.86027820010348985203998163708, −10.003707175892968243556322059801, −8.85425909691916766637954201686, −8.348911236022563793812286530883, −7.71796338166468655886785826896, −6.74129171302226954487264033943, −6.34570090309814096493030565797, −4.569147867622043167689414302, −3.86391022777782519714457406737, −2.84648008001918017507607213323, −1.510883726341225691316646767086, −0.44307979774837213927039911778, 1.620967709503771362216996590242, 2.1241784017278077618108648230, 3.46949690812925413761635443009, 3.921688811349722848652782703219, 4.923554773879229565062138384196, 6.66607136829908054783687925405, 7.409149954264149789366729650335, 8.42131800448859310356356316060, 8.92962073022289264138127278877, 9.3077998623873723705934692016, 10.63263498525848877940782966189, 11.26879159008543422923987354140, 12.3201855804261377171058152331, 12.71170184947458632964797836659, 13.75195104302821932661985023862, 15.02185254419044193855212965336, 15.4653001231002380232527954203, 16.16504183908795983607909545875, 17.13841609597000373598276937080, 18.1749117169016793427529268317, 18.83477755469292901831758796836, 19.51209667197072108494237715093, 20.050143379424819923055084229998, 20.72160675757710881826593756866, 21.58523045256921245750468652802

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