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Properties

Label 1-1045-1045.274-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.910 - 0.412i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + 7-s − 8-s + (−0.5 + 0.866i)9-s + 12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + 7-s − 8-s + (−0.5 + 0.866i)9-s + 12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.910 - 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.431976594 - 0.3094889788i\)
\(L(\frac12)\) \(\approx\) \(1.431976594 - 0.3094889788i\)
\(L(1)\) \(\approx\) \(1.141605017 + 0.1169816302i\)
\(L(1)\) \(\approx\) \(1.141605017 + 0.1169816302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−21.53817457451771688673245523834, −21.015461737672645216998354049403, −20.318156879861495046701550580884, −19.46529547396300598755795467206, −18.48805707079205358081731582099, −17.74872394656718152047203145849, −17.02077788053780794119698632135, −16.01080578092229760102376873976, −14.99470777264187811253301685699, −14.689748572163352634704510849370, −13.65445632207657991439048322021, −12.8247221332148667613156782464, −11.67866379456754566345971563568, −11.291798479003068134807979875032, −10.731561898835989103825520275258, −9.66758982879013797074991578361, −9.06390661006571470250096320658, −8.07766208133148031973569876044, −6.53921973853462991064287646876, −5.73547921260261043681556962285, −4.872396919594017237958495606505, −4.19164489509656093070885559278, −3.47821548847252338097108318794, −2.14162807053687868534115424770, −1.19875604478965166483486360527, 0.617193512313434169591465638981, 2.0157767325544476710325072021, 3.11563841566709222825185448114, 4.41781057662778852841296393754, 5.266045238489267989428660616171, 5.802115208654606052995227683832, 6.9594739699743654026844790658, 7.415622276488379713682207364901, 8.366607770332024051015284709, 8.935015370462266225689099403999, 10.57376841213485289808353017856, 11.33201312756217586887430151919, 12.13220809653482344519621515702, 12.95101625843581927010360211970, 13.57099277024717063683771051733, 14.39382304800065461900753180791, 15.09994198470349651335060106894, 16.08292510129237296118101855638, 16.85648573854119715756318145726, 17.554053748805463863487688238379, 18.28971070781456144784361493755, 18.608699001699750355802303185590, 20.19717006086645698080853112160, 20.641477345985804436135381811903, 21.925081942240592764606675460529

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