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Properties

Label 1-1033-1033.124-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.985 + 0.170i$
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  + (−0.276 + 0.961i)2-s + (0.985 − 0.169i)3-s + (−0.847 − 0.531i)4-s + (0.999 − 0.0121i)5-s + (−0.109 + 0.994i)6-s + (0.694 − 0.719i)7-s + (0.744 − 0.667i)8-s + (0.942 − 0.334i)9-s + (−0.264 + 0.964i)10-s + (0.827 − 0.561i)11-s + (−0.925 − 0.379i)12-s + (−0.334 − 0.942i)13-s + (0.5 + 0.866i)14-s + (0.983 − 0.181i)15-s + (0.435 + 0.900i)16-s + (0.658 + 0.752i)17-s + ⋯
L(s)  = 1  + (−0.276 + 0.961i)2-s + (0.985 − 0.169i)3-s + (−0.847 − 0.531i)4-s + (0.999 − 0.0121i)5-s + (−0.109 + 0.994i)6-s + (0.694 − 0.719i)7-s + (0.744 − 0.667i)8-s + (0.942 − 0.334i)9-s + (−0.264 + 0.964i)10-s + (0.827 − 0.561i)11-s + (−0.925 − 0.379i)12-s + (−0.334 − 0.942i)13-s + (0.5 + 0.866i)14-s + (0.983 − 0.181i)15-s + (0.435 + 0.900i)16-s + (0.658 + 0.752i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.390144316 + 0.2049066171i\)
\(L(\frac12)\) \(\approx\) \(2.390144316 + 0.2049066171i\)
\(L(1)\) \(\approx\) \(1.556783788 + 0.2782867787i\)
\(L(1)\) \(\approx\) \(1.556783788 + 0.2782867787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.276 + 0.961i)T \)
3 \( 1 + (0.985 - 0.169i)T \)
5 \( 1 + (0.999 - 0.0121i)T \)
7 \( 1 + (0.694 - 0.719i)T \)
11 \( 1 + (0.827 - 0.561i)T \)
13 \( 1 + (-0.334 - 0.942i)T \)
17 \( 1 + (0.658 + 0.752i)T \)
19 \( 1 + (-0.541 - 0.840i)T \)
23 \( 1 + (-0.591 + 0.806i)T \)
29 \( 1 + (-0.685 + 0.728i)T \)
31 \( 1 + (-0.859 + 0.510i)T \)
37 \( 1 + (-0.997 - 0.0729i)T \)
41 \( 1 + (0.760 - 0.648i)T \)
43 \( 1 + (0.620 + 0.783i)T \)
47 \( 1 + (0.311 - 0.950i)T \)
53 \( 1 + (-0.489 + 0.872i)T \)
59 \( 1 + (0.685 + 0.728i)T \)
61 \( 1 + (-0.957 + 0.288i)T \)
67 \( 1 + (-0.998 + 0.0608i)T \)
71 \( 1 + (-0.999 - 0.0121i)T \)
73 \( 1 + (-0.0729 - 0.997i)T \)
79 \( 1 + (-0.900 - 0.435i)T \)
83 \( 1 + (0.950 - 0.311i)T \)
89 \( 1 + (-0.357 + 0.934i)T \)
97 \( 1 + (-0.987 - 0.157i)T \)
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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

−21.22837132254381486132538492544, −20.8185759524070864170171999667, −20.248307844510068615389970966859, −19.019221079196685095294462659146, −18.76665110390031204048485124491, −17.87269224126368454463864428756, −17.06088618997211938006814980383, −16.24692366039940337891296550716, −14.71009046782031535965082535253, −14.41255804512240708457075029185, −13.77969700522933591107067692938, −12.663745312536488572454597034914, −12.13113773341307000926937803378, −11.11526815662027524842891515547, −10.0614574301858609916320094430, −9.466995925792837786874164917144, −8.9782808851564936441978028208, −8.09340704145598389991837408063, −7.13409604677181069757706205066, −5.78747697834675210823759191137, −4.66972568673423498216149842958, −4.00780710977483034655616697824, −2.76674170456457476085460191998, −1.96231360107958606505061556843, −1.58989962387109326539540114374, 1.105940659889207764293406126220, 1.805343374855951570028785631, 3.30273080890347443550450005365, 4.20177240057817117515072290906, 5.29863050339827357418159328803, 6.10505867489722354069147514223, 7.162288720784352482710008874657, 7.69669848029676708798705577682, 8.74320426271360398030710123759, 9.158866237202050088729005932936, 10.23015572310721479138399944566, 10.73245738191597842764239143544, 12.45575357662428468047231632908, 13.32020150182753749805428832381, 13.862267326876090182940997047961, 14.598241856587688038720391884605, 14.97397670173078981575150412909, 16.163483577336918346493960042685, 17.04604724352540790813224870390, 17.62429843804082341457951332276, 18.22006410209598801245932427085, 19.341444694300007996640617354743, 19.804680381618159523213606223776, 20.7988256473933356622985398384, 21.65912515921553311948916870803

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