↑

Properties

Label 1-1049-1049.102-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.167 + 0.985i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.989 + 0.143i)2-s + (−0.260 + 0.965i)3-s + (0.958 + 0.283i)4-s + (−0.752 − 0.658i)5-s + (−0.396 + 0.918i)6-s + (−0.962 + 0.272i)7-s + (0.908 + 0.418i)8-s + (−0.864 − 0.503i)9-s + (−0.649 − 0.760i)10-s + (0.951 + 0.306i)11-s + (−0.524 + 0.851i)12-s + (0.719 − 0.694i)13-s + (−0.991 + 0.131i)14-s + (0.832 − 0.554i)15-s + (0.838 + 0.544i)16-s + (0.979 + 0.202i)17-s + ⋯
L(s)  = 1  + (0.989 + 0.143i)2-s + (−0.260 + 0.965i)3-s + (0.958 + 0.283i)4-s + (−0.752 − 0.658i)5-s + (−0.396 + 0.918i)6-s + (−0.962 + 0.272i)7-s + (0.908 + 0.418i)8-s + (−0.864 − 0.503i)9-s + (−0.649 − 0.760i)10-s + (0.951 + 0.306i)11-s + (−0.524 + 0.851i)12-s + (0.719 − 0.694i)13-s + (−0.991 + 0.131i)14-s + (0.832 − 0.554i)15-s + (0.838 + 0.544i)16-s + (0.979 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.167 + 0.985i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.167 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339070872 + 1.586006608i\)
\(L(\frac12)\) \(\approx\) \(1.339070872 + 1.586006608i\)
\(L(1)\) \(\approx\) \(1.380631425 + 0.6469153532i\)
\(L(1)\) \(\approx\) \(1.380631425 + 0.6469153532i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.989 + 0.143i)T \)
3 \( 1 + (-0.260 + 0.965i)T \)
5 \( 1 + (-0.752 - 0.658i)T \)
7 \( 1 + (-0.962 + 0.272i)T \)
11 \( 1 + (0.951 + 0.306i)T \)
13 \( 1 + (0.719 - 0.694i)T \)
17 \( 1 + (0.979 + 0.202i)T \)
19 \( 1 + (-0.736 + 0.676i)T \)
23 \( 1 + (-0.190 + 0.981i)T \)
29 \( 1 + (0.995 - 0.0957i)T \)
31 \( 1 + (-0.931 - 0.363i)T \)
37 \( 1 + (-0.711 + 0.702i)T \)
41 \( 1 + (0.676 + 0.736i)T \)
43 \( 1 + (-0.472 - 0.881i)T \)
47 \( 1 + (0.503 - 0.864i)T \)
53 \( 1 + (-0.767 + 0.640i)T \)
59 \( 1 + (-0.131 + 0.991i)T \)
61 \( 1 + (-0.340 + 0.940i)T \)
67 \( 1 + (0.374 + 0.927i)T \)
71 \( 1 + (0.396 + 0.918i)T \)
73 \( 1 + (-0.797 + 0.603i)T \)
79 \( 1 + (0.340 + 0.940i)T \)
83 \( 1 + (0.948 + 0.318i)T \)
89 \( 1 + (0.962 + 0.272i)T \)
97 \( 1 + (-0.935 - 0.352i)T \)
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

−21.60551813984165423278239927200, −20.41597289435871186109303166194, −19.56412716969564234203310621809, −19.20315942263704269460437588596, −18.59047634103989230249532327155, −17.31963706854540764971785464618, −16.29512444314762923098001897569, −16.0208283837612106329436701101, −14.626806968358671382741912523992, −14.202274559118699884257449696798, −13.4404655740865704993051384563, −12.37035509430941931622874049123, −12.163788209887184651270493894500, −11.031031469941998109679719562812, −10.70121461898313487784964913096, −9.23541268610416556258539752137, −8.06028446418824110880876555380, −7.089991234331475433958244500824, −6.503840707267436572939243103983, −6.117378978206559977829586651514, −4.68902160629343625837371171660, −3.65608191140114116867755128766, −3.078425540594205585044647532183, −1.956594308592584449130809268146, −0.6993605832212950883620867160, 1.2759444598507924455646991908, 2.969496585972984851209026074602, 3.778296296475806588179118496591, 4.10385462842488528612049781860, 5.37040194712801723958347769121, 5.86815272148321516781800307996, 6.836031867803411749145142750, 8.02543960623954754559767941383, 8.842356404757593566141571266606, 9.87632340789572770387841990807, 10.66290960378005214639929257431, 11.73532272808901445577530988800, 12.16739806337887578954234697278, 12.92144577656322389334766758726, 13.93580815683583176821939341246, 14.98503099669441962665013340620, 15.377068785345839961400205559, 16.20206342707510428673589280019, 16.6526520613221570455559237139, 17.39393707475739057561733970042, 18.95974627305017148761133857088, 19.79333145850562803091102616520, 20.30180518015426541433169438544, 21.098233050457918720997948437076, 21.83915983190589482200032147855

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