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Properties

Label 1-1033-1033.108-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.170 + 0.985i$
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  + (−0.847 − 0.531i)2-s + (−0.942 + 0.334i)3-s + (0.435 + 0.900i)4-s + (−0.0243 − 0.999i)5-s + (0.976 + 0.217i)6-s + (0.0365 + 0.999i)7-s + (0.109 − 0.994i)8-s + (0.776 − 0.630i)9-s + (−0.510 + 0.859i)10-s + (−0.929 − 0.368i)11-s + (−0.711 − 0.702i)12-s + (0.630 + 0.776i)13-s + (0.5 − 0.866i)14-s + (0.357 + 0.934i)15-s + (−0.620 + 0.783i)16-s + (0.133 − 0.991i)17-s + ⋯
L(s)  = 1  + (−0.847 − 0.531i)2-s + (−0.942 + 0.334i)3-s + (0.435 + 0.900i)4-s + (−0.0243 − 0.999i)5-s + (0.976 + 0.217i)6-s + (0.0365 + 0.999i)7-s + (0.109 − 0.994i)8-s + (0.776 − 0.630i)9-s + (−0.510 + 0.859i)10-s + (−0.929 − 0.368i)11-s + (−0.711 − 0.702i)12-s + (0.630 + 0.776i)13-s + (0.5 − 0.866i)14-s + (0.357 + 0.934i)15-s + (−0.620 + 0.783i)16-s + (0.133 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.170 + 0.985i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.170 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2892437027 + 0.2435941063i\)
\(L(\frac12)\) \(\approx\) \(0.2892437027 + 0.2435941063i\)
\(L(1)\) \(\approx\) \(0.4845173816 + 0.02812314194i\)
\(L(1)\) \(\approx\) \(0.4845173816 + 0.02812314194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.847 - 0.531i)T \)
3 \( 1 + (-0.942 + 0.334i)T \)
5 \( 1 + (-0.0243 - 0.999i)T \)
7 \( 1 + (0.0365 + 0.999i)T \)
11 \( 1 + (-0.929 - 0.368i)T \)
13 \( 1 + (0.630 + 0.776i)T \)
17 \( 1 + (0.133 - 0.991i)T \)
19 \( 1 + (-0.413 + 0.910i)T \)
23 \( 1 + (0.954 - 0.299i)T \)
29 \( 1 + (-0.998 + 0.0608i)T \)
31 \( 1 + (-0.478 + 0.877i)T \)
37 \( 1 + (0.989 + 0.145i)T \)
41 \( 1 + (-0.157 + 0.987i)T \)
43 \( 1 + (0.229 - 0.973i)T \)
47 \( 1 + (0.591 - 0.806i)T \)
53 \( 1 + (-0.853 + 0.520i)T \)
59 \( 1 + (0.998 + 0.0608i)T \)
61 \( 1 + (-0.833 + 0.551i)T \)
67 \( 1 + (0.121 + 0.992i)T \)
71 \( 1 + (0.0243 - 0.999i)T \)
73 \( 1 + (-0.145 - 0.989i)T \)
79 \( 1 + (0.783 - 0.620i)T \)
83 \( 1 + (-0.806 + 0.591i)T \)
89 \( 1 + (-0.667 + 0.744i)T \)
97 \( 1 + (0.311 - 0.950i)T \)
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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

−21.399952381593146486244194022248, −20.47120930472761676778105347707, −19.52471443210097205323853093568, −18.79104126832658351668522258786, −18.192857981311232144671807143286, −17.38792493641301989961092877274, −17.09536784288290831406577492279, −15.94978572646792272517126761287, −15.33881992732741560638121467233, −14.57192622664462075113392246841, −13.360698393967189510408218086828, −12.86250971695391408524802593993, −11.10795671356993236544606493111, −11.06250482785956107285083730842, −10.35854237655749378504313254815, −9.53907838318686140551354955043, −8.00450852861323500748914858847, −7.54633340355466493020787482336, −6.81034641419921929895914658434, −6.04362575873035600521861958875, −5.251847309792300985285244908365, −4.04560406480399257083933572389, −2.62307282370995963836572476665, −1.48924447850071614471177581152, −0.29172156170941856294176982422, 1.028108996173739706491252379339, 2.016916622750018375936099626143, 3.27515606975622648839066894392, 4.39194619344217794426057567102, 5.302433664755027189280206502929, 6.07099305395281297381861753087, 7.21231182178314998488784706929, 8.27959088375305756354638153109, 9.02549984911294405550958638538, 9.58648778029733208553368636644, 10.64395027391848332902496548219, 11.36480355965091636409830969983, 12.04220912575912401906555190285, 12.68310662122555962973333956658, 13.42118717769674670413092854817, 15.0576672145443001882703307325, 15.93911798099231803495468313920, 16.41886427117162446361171191120, 16.90926627239636167676422221620, 18.08332296930624599978619012786, 18.47317942930319674540228141010, 19.139636868479011469825481351758, 20.48355483141690541757635702436, 20.95950421757162332799577516609, 21.453931218085253324422764658474

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