↑

Properties

Label 1-5077-5077.1039-r0-0-0
Degree $1$
Conductor $5077$
Sign $0.989 - 0.146i$
Analytic cond. $23.5774$
Root an. cond. $23.5774$
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  + (0.0606 + 0.998i)2-s + (−0.780 + 0.625i)3-s + (−0.992 + 0.121i)4-s + (−0.741 + 0.670i)5-s + (−0.671 − 0.740i)6-s + (0.0358 − 0.999i)7-s + (−0.180 − 0.983i)8-s + (0.217 − 0.976i)9-s + (−0.714 − 0.699i)10-s + (−0.999 + 0.0173i)11-s + (0.698 − 0.715i)12-s + (0.684 − 0.729i)13-s + (0.999 − 0.0247i)14-s + (0.159 − 0.987i)15-s + (0.970 − 0.240i)16-s + (−0.925 − 0.378i)17-s + ⋯
L(s)  = 1  + (0.0606 + 0.998i)2-s + (−0.780 + 0.625i)3-s + (−0.992 + 0.121i)4-s + (−0.741 + 0.670i)5-s + (−0.671 − 0.740i)6-s + (0.0358 − 0.999i)7-s + (−0.180 − 0.983i)8-s + (0.217 − 0.976i)9-s + (−0.714 − 0.699i)10-s + (−0.999 + 0.0173i)11-s + (0.698 − 0.715i)12-s + (0.684 − 0.729i)13-s + (0.999 − 0.0247i)14-s + (0.159 − 0.987i)15-s + (0.970 − 0.240i)16-s + (−0.925 − 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ 0.989 - 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5244431877 - 0.03859307707i\)
\(L(\frac12)\) \(\approx\) \(0.5244431877 - 0.03859307707i\)
\(L(1)\) \(\approx\) \(0.5226107826 + 0.3169078304i\)
\(L(1)\) \(\approx\) \(0.5226107826 + 0.3169078304i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (0.0606 + 0.998i)T \)
3 \( 1 + (-0.780 + 0.625i)T \)
5 \( 1 + (-0.741 + 0.670i)T \)
7 \( 1 + (0.0358 - 0.999i)T \)
11 \( 1 + (-0.999 + 0.0173i)T \)
13 \( 1 + (0.684 - 0.729i)T \)
17 \( 1 + (-0.925 - 0.378i)T \)
19 \( 1 + (0.799 + 0.600i)T \)
23 \( 1 + (0.303 + 0.952i)T \)
29 \( 1 + (-0.195 - 0.980i)T \)
31 \( 1 + (0.904 + 0.426i)T \)
37 \( 1 + (0.749 - 0.661i)T \)
41 \( 1 + (0.279 + 0.960i)T \)
43 \( 1 + (0.227 - 0.973i)T \)
47 \( 1 + (-0.546 + 0.837i)T \)
53 \( 1 + (0.820 - 0.572i)T \)
59 \( 1 + (-0.664 + 0.747i)T \)
61 \( 1 + (-0.229 - 0.973i)T \)
67 \( 1 + (0.354 + 0.934i)T \)
71 \( 1 + (-0.671 - 0.740i)T \)
73 \( 1 + (-0.997 - 0.0667i)T \)
79 \( 1 + (-0.272 - 0.962i)T \)
83 \( 1 + (-0.205 + 0.978i)T \)
89 \( 1 + (0.996 - 0.0890i)T \)
97 \( 1 + (-0.970 - 0.242i)T \)
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

−18.3083771393209653735078280111, −17.64099856319034753581765197611, −16.79054120145938260761518390016, −16.07573759372034093829722553331, −15.52925558770706221647986812280, −14.676304620915289627788045723395, −13.51173636812222912865194962233, −13.2288068036107279783183628698, −12.50315879826305113644896889060, −12.05928317990771533927594531901, −11.25318501562799401202683341039, −11.11175388078293355831653829581, −10.135486901958763566319168608, −9.13479504546959115549890221246, −8.593348914579600881194586518253, −8.067018974543471348498249162480, −7.12259601978649002970882854864, −6.15293456294718473751168196128, −5.45052133622704560004337201949, −4.741578516864635208076757802330, −4.341145020204815746891104334370, −3.09802910471601671733292848622, −2.414607496789095748161801042272, −1.59656415823744765544436552134, −0.77313425963839671231166509573, 0.24939500268292688438454753374, 1.06255964541290513202588966564, 2.926225913643208487025688420852, 3.56846232642896012560177338346, 4.2488041682598517002520703492, 4.83808145881483358420801999248, 5.68690664587077527647476906435, 6.28222098602850134914600436653, 7.09075937898942657662653916737, 7.65593378928203179039313948923, 8.17032448595189126878107525024, 9.24300512095408646940725658558, 10.046395102160342657660227618089, 10.50084042179879931263018995644, 11.20402853166189478679097151012, 11.83287561923295177971742772119, 12.88613238880302812788341898027, 13.4278141648417841350454296993, 14.12217073913825417333933648689, 15.02542111073794580973266579466, 15.46755780550672467736631097789, 16.05128271405217442456422762852, 16.36542230121143823690140066464, 17.4368221740637698232502216125, 17.798744552048178265887719945895

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