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Properties

Label 1-1033-1033.1021-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.732 + 0.680i$
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  + (−0.229 − 0.973i)2-s + (0.915 + 0.402i)3-s + (−0.894 + 0.446i)4-s + (0.193 + 0.981i)5-s + (0.181 − 0.983i)6-s + (−0.957 + 0.288i)7-s + (0.639 + 0.768i)8-s + (0.676 + 0.736i)9-s + (0.910 − 0.413i)10-s + (−0.121 − 0.992i)11-s + (−0.998 + 0.0486i)12-s + (−0.736 + 0.676i)13-s + (0.5 + 0.866i)14-s + (−0.217 + 0.976i)15-s + (0.601 − 0.798i)16-s + (−0.478 − 0.877i)17-s + ⋯
L(s)  = 1  + (−0.229 − 0.973i)2-s + (0.915 + 0.402i)3-s + (−0.894 + 0.446i)4-s + (0.193 + 0.981i)5-s + (0.181 − 0.983i)6-s + (−0.957 + 0.288i)7-s + (0.639 + 0.768i)8-s + (0.676 + 0.736i)9-s + (0.910 − 0.413i)10-s + (−0.121 − 0.992i)11-s + (−0.998 + 0.0486i)12-s + (−0.736 + 0.676i)13-s + (0.5 + 0.866i)14-s + (−0.217 + 0.976i)15-s + (0.601 − 0.798i)16-s + (−0.478 − 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.732 + 0.680i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.732 + 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1800236725 + 0.4586229072i\)
\(L(\frac12)\) \(\approx\) \(0.1800236725 + 0.4586229072i\)
\(L(1)\) \(\approx\) \(0.8372097015 + 0.007880994424i\)
\(L(1)\) \(\approx\) \(0.8372097015 + 0.007880994424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.229 - 0.973i)T \)
3 \( 1 + (0.915 + 0.402i)T \)
5 \( 1 + (0.193 + 0.981i)T \)
7 \( 1 + (-0.957 + 0.288i)T \)
11 \( 1 + (-0.121 - 0.992i)T \)
13 \( 1 + (-0.736 + 0.676i)T \)
17 \( 1 + (-0.478 - 0.877i)T \)
19 \( 1 + (-0.964 - 0.264i)T \)
23 \( 1 + (-0.648 + 0.760i)T \)
29 \( 1 + (0.468 + 0.883i)T \)
31 \( 1 + (0.658 + 0.752i)T \)
37 \( 1 + (0.391 + 0.920i)T \)
41 \( 1 + (-0.299 - 0.954i)T \)
43 \( 1 + (0.276 - 0.961i)T \)
47 \( 1 + (-0.938 - 0.345i)T \)
53 \( 1 + (-0.946 - 0.322i)T \)
59 \( 1 + (-0.468 + 0.883i)T \)
61 \( 1 + (0.0365 - 0.999i)T \)
67 \( 1 + (-0.827 - 0.561i)T \)
71 \( 1 + (-0.193 + 0.981i)T \)
73 \( 1 + (-0.920 - 0.391i)T \)
79 \( 1 + (-0.798 + 0.601i)T \)
83 \( 1 + (-0.345 - 0.938i)T \)
89 \( 1 + (0.424 + 0.905i)T \)
97 \( 1 + (-0.571 - 0.820i)T \)
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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

−21.17933702738034792954102940685, −20.214761307315304069133802984816, −19.62589239976544027798471743463, −19.10551041583016464729660223113, −17.90145189891893518633258358907, −17.392735398233475355472267372339, −16.56336549175620729498018724904, −15.70178412369963372990266301942, −15.04816372379613488885762744173, −14.33598273175248361806732838676, −13.18341491788269511107467864903, −12.93534225926198688602115418210, −12.29726670724622920234597901248, −10.22725553353496815075917706515, −9.82415085749620857744060215680, −9.05630911802084152305686988960, −8.12173409459717553619378559746, −7.70017071993577705287686005605, −6.52545525399636513073975509516, −6.02163712992272739993288802434, −4.49763222739650368816698780646, −4.18463886251968115402841264076, −2.64319718436727372972454699574, −1.54422057286017678791700399116, −0.1867937228369913941964194432, 1.82173702589846257852823712516, 2.741811662491022746481877509194, 3.13756551052263043282092918812, 4.05710604634955360383951029882, 5.134606056397833372801984009074, 6.49999509554462348044145021389, 7.35656746095048401461459536061, 8.499192381557352620890517806623, 9.14419071416420809040342526752, 9.90766967190058525317735139050, 10.48533668265415676505389481986, 11.37067309843342158152862213427, 12.274104977248180523725756193173, 13.38840971901168180857579726188, 13.807018955675120042675558974, 14.53715768630263555734088185021, 15.575803516761343471173555025759, 16.314790212810362458430147830222, 17.355441864140904671992255218039, 18.40420929145793417858497053662, 19.02807715325108784341528520775, 19.424622922560652925013711804554, 20.14174893232813962634825647363, 21.23497509850861503710708303381, 21.89291743720828862799145349823

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