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Properties

Label 1-1033-1033.75-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.998 - 0.0505i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.620 − 0.783i)2-s + (0.205 + 0.978i)3-s + (−0.229 + 0.973i)4-s + (0.995 + 0.0972i)5-s + (0.639 − 0.768i)6-s + (0.989 + 0.145i)7-s + (0.905 − 0.424i)8-s + (−0.915 + 0.402i)9-s + (−0.541 − 0.840i)10-s + (0.0608 − 0.998i)11-s + (−0.999 − 0.0243i)12-s + (−0.915 + 0.402i)13-s + (−0.5 − 0.866i)14-s + (0.109 + 0.994i)15-s + (−0.894 − 0.446i)16-s + (0.859 − 0.510i)17-s + ⋯
L(s)  = 1  + (−0.620 − 0.783i)2-s + (0.205 + 0.978i)3-s + (−0.229 + 0.973i)4-s + (0.995 + 0.0972i)5-s + (0.639 − 0.768i)6-s + (0.989 + 0.145i)7-s + (0.905 − 0.424i)8-s + (−0.915 + 0.402i)9-s + (−0.541 − 0.840i)10-s + (0.0608 − 0.998i)11-s + (−0.999 − 0.0243i)12-s + (−0.915 + 0.402i)13-s + (−0.5 − 0.866i)14-s + (0.109 + 0.994i)15-s + (−0.894 − 0.446i)16-s + (0.859 − 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.998 - 0.0505i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.998 - 0.0505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552837487 - 0.03927492112i\)
\(L(\frac12)\) \(\approx\) \(1.552837487 - 0.03927492112i\)
\(L(1)\) \(\approx\) \(1.093478709 - 0.03200017411i\)
\(L(1)\) \(\approx\) \(1.093478709 - 0.03200017411i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.620 - 0.783i)T \)
3 \( 1 + (0.205 + 0.978i)T \)
5 \( 1 + (0.995 + 0.0972i)T \)
7 \( 1 + (0.989 + 0.145i)T \)
11 \( 1 + (0.0608 - 0.998i)T \)
13 \( 1 + (-0.915 + 0.402i)T \)
17 \( 1 + (0.859 - 0.510i)T \)
19 \( 1 + (-0.133 - 0.991i)T \)
23 \( 1 + (0.345 + 0.938i)T \)
29 \( 1 + (0.970 + 0.241i)T \)
31 \( 1 + (-0.413 - 0.910i)T \)
37 \( 1 + (0.833 - 0.551i)T \)
41 \( 1 + (0.806 - 0.591i)T \)
43 \( 1 + (0.601 - 0.798i)T \)
47 \( 1 + (-0.820 - 0.571i)T \)
53 \( 1 + (-0.581 + 0.813i)T \)
59 \( 1 + (0.970 - 0.241i)T \)
61 \( 1 + (-0.694 + 0.719i)T \)
67 \( 1 + (0.883 + 0.468i)T \)
71 \( 1 + (0.995 - 0.0972i)T \)
73 \( 1 + (0.833 + 0.551i)T \)
79 \( 1 + (-0.894 + 0.446i)T \)
83 \( 1 + (-0.820 + 0.571i)T \)
89 \( 1 + (-0.976 - 0.217i)T \)
97 \( 1 + (0.299 - 0.954i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.45869413891335247703508273004, −20.58049163688622315069180072528, −19.910850320825860202146746962761, −19.00903017483248307386747585538, −18.15517551410639902867409064287, −17.713666548786356756469246649060, −17.15964207769984638951131860229, −16.454165044318382705023840110804, −14.88953127929553552414986413775, −14.53759240156213466614375494586, −14.063655909256815650479376903045, −12.83681265331740934623463213196, −12.32960539027980159628187789297, −11.00835507424836380471154860701, −10.08968069821477742168252183830, −9.47588377690483254936062026467, −8.26729523240187898962766166574, −7.9254100767720765896650246929, −6.956526880905859212780006543202, −6.22246169048573227448551050723, −5.32437333182905063488183874710, −4.58623708968178983499532170580, −2.65007891122795956293495137153, −1.7416998527386794275471556418, −1.1094723143062742120100374305, 0.98099991061362131547711799605, 2.29742220859275524257611503963, 2.82171611282145510592685177042, 3.98908378900902918395932383725, 4.97951842138953142320166350918, 5.62880609198456628118508609383, 7.1515359047735455110495915138, 8.121177375204425411350624816272, 9.01954454384518092415072923982, 9.47677156529126395126531619050, 10.28922468305943980838417901835, 11.164816410445805911045435282899, 11.54298918609717920138936343083, 12.75938462511408094327214737233, 13.9763539183486845327593224490, 14.1314456555948758872339640771, 15.31005763359796410435842699036, 16.380699528137071539421339855075, 17.05398703779731855710152278312, 17.56592170232516495747326514962, 18.464390509973678269351902060411, 19.33113589962562480689598726757, 20.10253421904128927948463044656, 21.01655068762918465991145258110, 21.48352632920072816659440519862

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