↑

Properties

Label 1-1049-1049.128-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.986 + 0.162i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.702 − 0.711i)2-s + (0.992 − 0.119i)3-s + (−0.0119 + 0.999i)4-s + (0.685 + 0.727i)5-s + (−0.782 − 0.622i)6-s + (0.667 + 0.744i)7-s + (0.719 − 0.694i)8-s + (0.971 − 0.237i)9-s + (0.0359 − 0.999i)10-s + (0.989 + 0.143i)11-s + (0.107 + 0.994i)12-s + (−0.472 + 0.881i)13-s + (0.0599 − 0.998i)14-s + (0.767 + 0.640i)15-s + (−0.999 − 0.0239i)16-s + (−0.944 − 0.329i)17-s + ⋯
L(s)  = 1  + (−0.702 − 0.711i)2-s + (0.992 − 0.119i)3-s + (−0.0119 + 0.999i)4-s + (0.685 + 0.727i)5-s + (−0.782 − 0.622i)6-s + (0.667 + 0.744i)7-s + (0.719 − 0.694i)8-s + (0.971 − 0.237i)9-s + (0.0359 − 0.999i)10-s + (0.989 + 0.143i)11-s + (0.107 + 0.994i)12-s + (−0.472 + 0.881i)13-s + (0.0599 − 0.998i)14-s + (0.767 + 0.640i)15-s + (−0.999 − 0.0239i)16-s + (−0.944 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.986 + 0.162i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.024753869 + 0.1655795327i\)
\(L(\frac12)\) \(\approx\) \(2.024753869 + 0.1655795327i\)
\(L(1)\) \(\approx\) \(1.353703374 - 0.06869315897i\)
\(L(1)\) \(\approx\) \(1.353703374 - 0.06869315897i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.702 - 0.711i)T \)
3 \( 1 + (0.992 - 0.119i)T \)
5 \( 1 + (0.685 + 0.727i)T \)
7 \( 1 + (0.667 + 0.744i)T \)
11 \( 1 + (0.989 + 0.143i)T \)
13 \( 1 + (-0.472 + 0.881i)T \)
17 \( 1 + (-0.944 - 0.329i)T \)
19 \( 1 + (0.811 + 0.583i)T \)
23 \( 1 + (-0.493 - 0.869i)T \)
29 \( 1 + (0.864 - 0.503i)T \)
31 \( 1 + (0.951 - 0.306i)T \)
37 \( 1 + (0.935 - 0.352i)T \)
41 \( 1 + (-0.811 - 0.583i)T \)
43 \( 1 + (-0.340 - 0.940i)T \)
47 \( 1 + (-0.971 + 0.237i)T \)
53 \( 1 + (-0.631 - 0.775i)T \)
59 \( 1 + (-0.0599 - 0.998i)T \)
61 \( 1 + (-0.429 - 0.903i)T \)
67 \( 1 + (0.513 + 0.857i)T \)
71 \( 1 + (-0.782 + 0.622i)T \)
73 \( 1 + (0.407 + 0.913i)T \)
79 \( 1 + (-0.429 + 0.903i)T \)
83 \( 1 + (-0.838 + 0.544i)T \)
89 \( 1 + (0.667 - 0.744i)T \)
97 \( 1 + (-0.918 - 0.396i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.427149005253791860659263002316, −20.275023783528941496730363963248, −19.96145338080103981123638131625, −19.4805585252646385343748316665, −17.91853990643735460407875945624, −17.803276394860010266579190116907, −16.85868596503935120144922509558, −16.108084015654123240993653684736, −15.18916696368840632043403302177, −14.55077804689180005077261059345, −13.61260851351007443081225432392, −13.42820496268648945468339916196, −11.94623775229716523323035449428, −10.785031163577282264350026085208, −9.913174585411802547003910639883, −9.40152775604771088328589961617, −8.5024466607804057806422087564, −7.98046378569504181082578322233, −7.053132376837587867211066149142, −6.167064147892247892237375539149, −4.89218668654331230529342512933, −4.447723255176754256055653183510, −2.95213867131071322558793420955, −1.64111340758987823643831178430, −1.10430190393312976332459144232, 1.43577337843204910797734372719, 2.16420261576680231331433467285, 2.7051868299509536492255960497, 3.85266742529196904469856409258, 4.731802058860367596683307660664, 6.42422017319712429605158442241, 7.01254352197823586389972283513, 8.103592650398935228292855485769, 8.765542611382983692803881542670, 9.61627114227414903045226301876, 9.97802073945634011900028110744, 11.27443279393489138367006293553, 11.85330991836529787482597981679, 12.72942924973737851195272715131, 13.95882521545037539963101715897, 14.152951266252613845176552780543, 15.16509344208041014803737760880, 16.09644844095796562854644168043, 17.25826529342421138454960502016, 17.860831890065796525334394430587, 18.67249010422303560939468193744, 19.03021172438757105944825796183, 20.03307775412121864122738663547, 20.63195439697663282982277859084, 21.51576773794848685134302635679

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