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Properties

Label 1-1049-1049.22-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.857 + 0.514i$
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  + (−0.513 − 0.857i)2-s + (0.979 − 0.202i)3-s + (−0.472 + 0.881i)4-s + (0.429 − 0.903i)5-s + (−0.676 − 0.736i)6-s + (−0.143 − 0.989i)7-s + (0.998 − 0.0479i)8-s + (0.918 − 0.396i)9-s + (−0.995 + 0.0957i)10-s + (−0.927 + 0.374i)11-s + (−0.283 + 0.958i)12-s + (−0.965 + 0.260i)13-s + (−0.775 + 0.631i)14-s + (0.237 − 0.971i)15-s + (−0.554 − 0.832i)16-s + (−0.931 + 0.363i)17-s + ⋯
L(s)  = 1  + (−0.513 − 0.857i)2-s + (0.979 − 0.202i)3-s + (−0.472 + 0.881i)4-s + (0.429 − 0.903i)5-s + (−0.676 − 0.736i)6-s + (−0.143 − 0.989i)7-s + (0.998 − 0.0479i)8-s + (0.918 − 0.396i)9-s + (−0.995 + 0.0957i)10-s + (−0.927 + 0.374i)11-s + (−0.283 + 0.958i)12-s + (−0.965 + 0.260i)13-s + (−0.775 + 0.631i)14-s + (0.237 − 0.971i)15-s + (−0.554 − 0.832i)16-s + (−0.931 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.857 + 0.514i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2327288115 - 0.8399274690i\)
\(L(\frac12)\) \(\approx\) \(-0.2327288115 - 0.8399274690i\)
\(L(1)\) \(\approx\) \(0.6498517198 - 0.6375200283i\)
\(L(1)\) \(\approx\) \(0.6498517198 - 0.6375200283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.513 - 0.857i)T \)
3 \( 1 + (0.979 - 0.202i)T \)
5 \( 1 + (0.429 - 0.903i)T \)
7 \( 1 + (-0.143 - 0.989i)T \)
11 \( 1 + (-0.927 + 0.374i)T \)
13 \( 1 + (-0.965 + 0.260i)T \)
17 \( 1 + (-0.931 + 0.363i)T \)
19 \( 1 + (0.908 + 0.418i)T \)
23 \( 1 + (-0.658 - 0.752i)T \)
29 \( 1 + (-0.935 - 0.352i)T \)
31 \( 1 + (-0.214 - 0.976i)T \)
37 \( 1 + (-0.818 + 0.574i)T \)
41 \( 1 + (-0.418 + 0.908i)T \)
43 \( 1 + (-0.992 + 0.119i)T \)
47 \( 1 + (-0.396 - 0.918i)T \)
53 \( 1 + (-0.249 + 0.968i)T \)
59 \( 1 + (0.631 - 0.775i)T \)
61 \( 1 + (0.612 - 0.790i)T \)
67 \( 1 + (0.991 - 0.131i)T \)
71 \( 1 + (0.676 - 0.736i)T \)
73 \( 1 + (-0.997 - 0.0718i)T \)
79 \( 1 + (-0.612 - 0.790i)T \)
83 \( 1 + (0.999 - 0.0359i)T \)
89 \( 1 + (0.143 - 0.989i)T \)
97 \( 1 + (0.534 + 0.845i)T \)
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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

−22.11130802341774953705966775985, −21.3497091783025224482400417087, −20.20730317607327709603389328458, −19.41754006637560716641220340834, −18.80046654211158105359111128780, −18.06014470875088626982632965406, −17.63318037021316992768903867597, −16.13256672730413959994628345470, −15.67594046916816738917519294127, −15.03601022338682657696467058387, −14.3240725762576690333394795052, −13.62508500642763890523938101461, −12.875831942864212324838500312528, −11.42896023522463830723586869469, −10.42294118229138982998805766801, −9.75346428389320415892323808007, −9.10830185112105535412377066596, −8.28879272745305120710520883044, −7.36151259938015519596231489310, −6.85219611940639654512582841770, −5.521777422707911578328879279129, −5.096556238645207141950385866299, −3.5265788219906936598534358185, −2.57283722598458866598825830980, −1.87978115053488939359674173623, 0.34348035033249310500042435608, 1.70287887302613536672219817311, 2.21907012439236407053085621606, 3.39350385996779408705779012917, 4.308295485319060533798916372359, 4.98404379642752885616317881692, 6.703289176401799998804917076322, 7.70354698655062758935367887228, 8.13770166417101015785708421627, 9.15463996974120140969548550129, 9.91496865453539554916532181237, 10.231728588414732971620297904819, 11.5833029267565287626555718671, 12.53348383231414070471904956230, 13.17787954677993180722540713174, 13.59124869258806479750417616741, 14.5362719350687671341481647752, 15.729538872808822369891677261587, 16.615422844515521640992543055811, 17.27331213456343310275971795102, 18.13133519855316980812763632420, 18.84454997657127624096076527471, 19.841260460599155976053508749182, 20.374746310615159181529721957006, 20.55563003749138111584132446377

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