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Properties

Label 1-1033-1033.130-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.728 + 0.685i$
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  + (0.744 + 0.667i)2-s + (0.872 + 0.489i)3-s + (0.109 + 0.994i)4-s + (0.999 + 0.0365i)5-s + (0.322 + 0.946i)6-s + (−0.744 + 0.667i)7-s + (−0.581 + 0.813i)8-s + (0.520 + 0.853i)9-s + (0.719 + 0.694i)10-s + (−0.217 + 0.976i)11-s + (−0.391 + 0.920i)12-s + (0.853 − 0.520i)13-s − 14-s + (0.853 + 0.520i)15-s + (−0.976 + 0.217i)16-s + (−0.833 − 0.551i)17-s + ⋯
L(s)  = 1  + (0.744 + 0.667i)2-s + (0.872 + 0.489i)3-s + (0.109 + 0.994i)4-s + (0.999 + 0.0365i)5-s + (0.322 + 0.946i)6-s + (−0.744 + 0.667i)7-s + (−0.581 + 0.813i)8-s + (0.520 + 0.853i)9-s + (0.719 + 0.694i)10-s + (−0.217 + 0.976i)11-s + (−0.391 + 0.920i)12-s + (0.853 − 0.520i)13-s − 14-s + (0.853 + 0.520i)15-s + (−0.976 + 0.217i)16-s + (−0.833 − 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.728 + 0.685i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.728 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.285612224 + 3.240580772i\)
\(L(\frac12)\) \(\approx\) \(1.285612224 + 3.240580772i\)
\(L(1)\) \(\approx\) \(1.633876637 + 1.549455323i\)
\(L(1)\) \(\approx\) \(1.633876637 + 1.549455323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.744 + 0.667i)T \)
3 \( 1 + (0.872 + 0.489i)T \)
5 \( 1 + (0.999 + 0.0365i)T \)
7 \( 1 + (-0.744 + 0.667i)T \)
11 \( 1 + (-0.217 + 0.976i)T \)
13 \( 1 + (0.853 - 0.520i)T \)
17 \( 1 + (-0.833 - 0.551i)T \)
19 \( 1 + (0.989 + 0.145i)T \)
23 \( 1 + (0.946 - 0.322i)T \)
29 \( 1 + (0.768 - 0.639i)T \)
31 \( 1 + (0.0365 - 0.999i)T \)
37 \( 1 + (-0.976 + 0.217i)T \)
41 \( 1 + (-0.520 + 0.853i)T \)
43 \( 1 + (-0.905 - 0.424i)T \)
47 \( 1 + (-0.813 - 0.581i)T \)
53 \( 1 + (0.999 + 0.0365i)T \)
59 \( 1 + (-0.768 - 0.639i)T \)
61 \( 1 + (-0.639 - 0.768i)T \)
67 \( 1 + (-0.983 - 0.181i)T \)
71 \( 1 + (-0.999 + 0.0365i)T \)
73 \( 1 + (0.217 - 0.976i)T \)
79 \( 1 + (-0.217 + 0.976i)T \)
83 \( 1 + (0.581 + 0.813i)T \)
89 \( 1 + (0.889 + 0.457i)T \)
97 \( 1 + (-0.889 + 0.457i)T \)
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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

−21.27318022320567676476301371210, −20.56068882068841361248098347812, −19.771509990767949456319806163637, −19.19733363291293567735602824698, −18.40118832324182737411469620509, −17.6613308258771264593298766620, −16.35083253139005860872415851918, −15.67505888184724673230647954126, −14.56109132199393375038579994053, −13.74566605659522947110478062080, −13.51096697496424873154009768588, −12.94135725878587302818239422052, −11.90013670787882665610448298604, −10.75045303027427721959812364186, −10.21824717333060493959456833129, −9.09527240700961509436638353712, −8.76053871487314044841251624712, −7.069718306712619574742834099099, −6.52430138875393358388897559295, −5.69521825139165027354538883120, −4.51694074765252450525351626267, −3.27456098351556766926535956271, −3.07999077419286051115122792497, −1.71468634593486654803919113625, −1.066351636335138586577505269524, 1.93596121249718192181531099660, 2.78521622649728097090052925714, 3.400156807629519597785141305329, 4.694305355402591026815297368256, 5.28140260108633438975176129907, 6.326442087320653589230207244608, 7.039021653203562438739767067471, 8.14275699969573225618467855171, 8.97738787472364601782565196348, 9.60436246619663004604013439196, 10.464854002151661928887252701817, 11.741101238335666597418756848136, 12.81434247559202348547312343663, 13.40061873547210098428982344015, 13.83084248106205196619846367802, 15.04599058238443060564801129794, 15.32008632261048250953301875439, 16.12784759415496662394232031181, 16.935533978831420253671407216483, 18.02542134241788966202097150168, 18.49027218486004569994359365326, 19.86276426704977134500266046097, 20.63335335116006155390279673607, 21.05914757145749342704536406772, 22.0134454160020285718501153006

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