↑

Properties

Label 1-1021-1021.33-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.999 + 0.00464i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (−0.687 + 0.726i)2-s + (−0.273 − 0.961i)3-s + (−0.0554 − 0.998i)4-s + (0.843 − 0.536i)5-s + (0.886 + 0.462i)6-s + (0.659 + 0.751i)7-s + (0.763 + 0.645i)8-s + (−0.850 + 0.526i)9-s + (−0.189 + 0.981i)10-s + (−0.996 + 0.0861i)11-s + (−0.945 + 0.326i)12-s + (0.982 + 0.183i)13-s + (−0.999 − 0.0369i)14-s + (−0.747 − 0.664i)15-s + (−0.993 + 0.110i)16-s + (−0.622 − 0.782i)17-s + ⋯
L(s)  = 1  + (−0.687 + 0.726i)2-s + (−0.273 − 0.961i)3-s + (−0.0554 − 0.998i)4-s + (0.843 − 0.536i)5-s + (0.886 + 0.462i)6-s + (0.659 + 0.751i)7-s + (0.763 + 0.645i)8-s + (−0.850 + 0.526i)9-s + (−0.189 + 0.981i)10-s + (−0.996 + 0.0861i)11-s + (−0.945 + 0.326i)12-s + (0.982 + 0.183i)13-s + (−0.999 − 0.0369i)14-s + (−0.747 − 0.664i)15-s + (−0.993 + 0.110i)16-s + (−0.622 − 0.782i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.999 + 0.00464i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.999 + 0.00464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110578351 + 0.002580757535i\)
\(L(\frac12)\) \(\approx\) \(1.110578351 + 0.002580757535i\)
\(L(1)\) \(\approx\) \(0.8467319720 + 0.003991985086i\)
\(L(1)\) \(\approx\) \(0.8467319720 + 0.003991985086i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.687 + 0.726i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (0.843 - 0.536i)T \)
7 \( 1 + (0.659 + 0.751i)T \)
11 \( 1 + (-0.996 + 0.0861i)T \)
13 \( 1 + (0.982 + 0.183i)T \)
17 \( 1 + (-0.622 - 0.782i)T \)
19 \( 1 + (-0.881 + 0.473i)T \)
23 \( 1 + (0.401 + 0.916i)T \)
29 \( 1 + (0.980 + 0.195i)T \)
31 \( 1 + (0.434 + 0.900i)T \)
37 \( 1 + (-0.0677 - 0.997i)T \)
41 \( 1 + (0.881 - 0.473i)T \)
43 \( 1 + (-0.489 - 0.872i)T \)
47 \( 1 + (0.467 + 0.883i)T \)
53 \( 1 + (0.862 + 0.505i)T \)
59 \( 1 + (0.641 + 0.767i)T \)
61 \( 1 + (0.531 - 0.846i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.763 - 0.645i)T \)
73 \( 1 + (-0.999 + 0.0369i)T \)
79 \( 1 + (0.956 + 0.291i)T \)
83 \( 1 + (-0.696 + 0.717i)T \)
89 \( 1 + (-0.153 - 0.988i)T \)
97 \( 1 + (0.908 + 0.417i)T \)
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

−21.33979440508253348536873420758, −20.90754477844836302596430044685, −20.34034465180393789759202370555, −19.28022633603329182437046144109, −18.28603149770255010657856759558, −17.69523324290108556220798609975, −17.13766974779482722292690037005, −16.3621469466457123605490903097, −15.39410506572507674529617669533, −14.57608769658340980128183836441, −13.4313155607361380327358598289, −13.036886960006154708941699525554, −11.51214562407179437629612865841, −10.9455148023083453379468950972, −10.374534501320675658139824031252, −9.96099626665672704819667449836, −8.63356965431709240307373537149, −8.30210841221385197059292650793, −6.89131777594261382815914997440, −6.03967738508287230323621341625, −4.77404078827827164277212584231, −4.08764938664709420661442832922, −2.98089698623379870998844772549, −2.1846472473772771154176221457, −0.83137331924533433775793643247, 0.90780856609204126841632104816, 1.85157167683352449396497897338, 2.51926355043205700591227522010, 4.716004280915399547025782799435, 5.45668285987529062745525308562, 6.00565345153879649822407049264, 6.90493420855647799632411249938, 7.86586387265558721338555231965, 8.66019719267335743370308238950, 9.056763806978663511525226699248, 10.41393034639743829885170281788, 11.04575388405901865873640339714, 12.09574423933434723989373239807, 13.04355381373472619824766108514, 13.75759708411045956957677140254, 14.37662563364000194135667729730, 15.60299708622331769855520171474, 16.128923419335426489148746692864, 17.21987910940732326984954152797, 17.77279368533214870289178443119, 18.24032677162106922299193963524, 18.89044740413092413180060309053, 19.826424761458406934468151828518, 20.836590636128013957277587378558, 21.417871517106163470075717618305

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