↑

Properties

Label 1-1049-1049.1023-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.304 + 0.952i$
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  + (−0.918 − 0.396i)2-s + (0.295 + 0.955i)3-s + (0.685 + 0.727i)4-s + (0.450 − 0.892i)5-s + (0.107 − 0.994i)6-s + (−0.864 − 0.503i)7-s + (−0.340 − 0.940i)8-s + (−0.825 + 0.564i)9-s + (−0.767 + 0.640i)10-s + (−0.935 − 0.352i)11-s + (−0.493 + 0.869i)12-s + (−0.429 + 0.903i)13-s + (0.593 + 0.804i)14-s + (0.985 + 0.167i)15-s + (−0.0599 + 0.998i)16-s + (0.667 + 0.744i)17-s + ⋯
L(s)  = 1  + (−0.918 − 0.396i)2-s + (0.295 + 0.955i)3-s + (0.685 + 0.727i)4-s + (0.450 − 0.892i)5-s + (0.107 − 0.994i)6-s + (−0.864 − 0.503i)7-s + (−0.340 − 0.940i)8-s + (−0.825 + 0.564i)9-s + (−0.767 + 0.640i)10-s + (−0.935 − 0.352i)11-s + (−0.493 + 0.869i)12-s + (−0.429 + 0.903i)13-s + (0.593 + 0.804i)14-s + (0.985 + 0.167i)15-s + (−0.0599 + 0.998i)16-s + (0.667 + 0.744i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.304 + 0.952i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.304 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3010433320 + 0.4122881416i\)
\(L(\frac12)\) \(\approx\) \(0.3010433320 + 0.4122881416i\)
\(L(1)\) \(\approx\) \(0.6256065755 + 0.06806808736i\)
\(L(1)\) \(\approx\) \(0.6256065755 + 0.06806808736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.918 - 0.396i)T \)
3 \( 1 + (0.295 + 0.955i)T \)
5 \( 1 + (0.450 - 0.892i)T \)
7 \( 1 + (-0.864 - 0.503i)T \)
11 \( 1 + (-0.935 - 0.352i)T \)
13 \( 1 + (-0.429 + 0.903i)T \)
17 \( 1 + (0.667 + 0.744i)T \)
19 \( 1 + (-0.0119 - 0.999i)T \)
23 \( 1 + (0.875 - 0.482i)T \)
29 \( 1 + (-0.249 + 0.968i)T \)
31 \( 1 + (0.702 + 0.711i)T \)
37 \( 1 + (-0.782 - 0.622i)T \)
41 \( 1 + (0.0119 + 0.999i)T \)
43 \( 1 + (0.0838 + 0.996i)T \)
47 \( 1 + (0.825 - 0.564i)T \)
53 \( 1 + (-0.797 - 0.603i)T \)
59 \( 1 + (-0.593 + 0.804i)T \)
61 \( 1 + (0.318 + 0.948i)T \)
67 \( 1 + (-0.534 - 0.845i)T \)
71 \( 1 + (0.107 + 0.994i)T \)
73 \( 1 + (-0.965 + 0.260i)T \)
79 \( 1 + (0.318 - 0.948i)T \)
83 \( 1 + (-0.131 + 0.991i)T \)
89 \( 1 + (-0.864 + 0.503i)T \)
97 \( 1 + (-0.851 + 0.524i)T \)
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

−20.87534904802774710075042064898, −20.47275424020109220679084065118, −19.15168492853035743825609192268, −18.9371805724179545048009903973, −18.37938743151822754810792536678, −17.49915807263785756015302298511, −16.979020106130405546293318623512, −15.56393532426621746362055960638, −15.286685926760367668971851189753, −14.25991470452286402957680934047, −13.51891279116447097554767328542, −12.53602981943404676471194928764, −11.78863789550746420369609048980, −10.649874103591409767932050048393, −9.91873718330363798101424669476, −9.31432151665549247828559274507, −8.123629267006176732125029598776, −7.49785610231922120605918835537, −6.87422965832284323193695194173, −5.85754450773704430934432434172, −5.48788248021386196604937154854, −3.13019952767236090135473496343, −2.69404468369099338127678629107, −1.76932916604024733832625816211, −0.290697114247115057573363141797, 1.14396272876063482372708986549, 2.48832660518235385306921915079, 3.216980619793319315912210235868, 4.28014514374183562257113056532, 5.18072733434475912287151470459, 6.33622229561506395589422487539, 7.377654236196067941191377161949, 8.45517672875951436261651976876, 9.01535647579120941570211055046, 9.71427971183241058831712497002, 10.392688133878627683856385620378, 11.03463455125356514277735353225, 12.22344970534066415523720905498, 12.99213183368423251506795179240, 13.72996915377250494665921805705, 14.90435066628982984098582843612, 15.97558015213460461407885653049, 16.3511089334814817725007847123, 16.94944458338010354561605694186, 17.67508477890803436463996797632, 18.91726463475281992564218677128, 19.55635131191078431228785925852, 20.12976422136952696782022608124, 21.08459867201935542106916320153, 21.34503365659639203958925854015

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