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Properties

Label 1-39e2-1521.241-r1-0-0
Degree $1$
Conductor $1521$
Sign $0.645 - 0.763i$
Analytic cond. $163.454$
Root an. cond. $163.454$
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  + (−0.160 + 0.987i)2-s + (−0.948 − 0.316i)4-s + (0.391 − 0.919i)5-s + (0.464 − 0.885i)7-s + (0.464 − 0.885i)8-s + (0.845 + 0.534i)10-s + (−0.774 + 0.632i)11-s + (0.799 + 0.600i)14-s + (0.799 + 0.600i)16-s + (0.845 − 0.534i)17-s + (−0.866 − 0.5i)19-s + (−0.663 + 0.748i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.692 − 0.721i)25-s + ⋯
L(s)  = 1  + (−0.160 + 0.987i)2-s + (−0.948 − 0.316i)4-s + (0.391 − 0.919i)5-s + (0.464 − 0.885i)7-s + (0.464 − 0.885i)8-s + (0.845 + 0.534i)10-s + (−0.774 + 0.632i)11-s + (0.799 + 0.600i)14-s + (0.799 + 0.600i)16-s + (0.845 − 0.534i)17-s + (−0.866 − 0.5i)19-s + (−0.663 + 0.748i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.692 − 0.721i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.645 - 0.763i$
Analytic conductor: \(163.454\)
Root analytic conductor: \(163.454\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1521,\ (1:\ ),\ 0.645 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.655440148 - 0.7677856457i\)
\(L(\frac12)\) \(\approx\) \(1.655440148 - 0.7677856457i\)
\(L(1)\) \(\approx\) \(1.008431085 + 0.08742177985i\)
\(L(1)\) \(\approx\) \(1.008431085 + 0.08742177985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.160 + 0.987i)T \)
5 \( 1 + (0.391 - 0.919i)T \)
7 \( 1 + (0.464 - 0.885i)T \)
11 \( 1 + (-0.774 + 0.632i)T \)
17 \( 1 + (0.845 - 0.534i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (0.987 + 0.160i)T \)
31 \( 1 + (0.721 + 0.692i)T \)
37 \( 1 + (0.960 - 0.278i)T \)
41 \( 1 + (0.822 + 0.568i)T \)
43 \( 1 + (0.970 - 0.239i)T \)
47 \( 1 + (0.979 + 0.200i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (-0.600 - 0.799i)T \)
61 \( 1 + (0.885 - 0.464i)T \)
67 \( 1 + (0.663 - 0.748i)T \)
71 \( 1 + (-0.903 + 0.428i)T \)
73 \( 1 + (0.935 + 0.354i)T \)
79 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (0.903 + 0.428i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.992 + 0.120i)T \)
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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

−20.84618851718159664772383640850, −19.51926865710786038754937672214, −18.97576290366418108415184907345, −18.43309136998400904591871344535, −17.8234543848850780735212327152, −17.0804900017197976786474261908, −16.00895338843637475307594617118, −15.00443898579210985680605338954, −14.36496847483529179063815478487, −13.67927209566994965236166006729, −12.79257957072933629893657743999, −12.008358737860789345250939654444, −11.342728364792989048077351742395, −10.43499425410516615299305615728, −10.12530236357049414147819342059, −9.034281064353997015469549169070, −8.19731952768902812237640125218, −7.657573826078625240291193822607, −6.02799213010121546628658821911, −5.73679964962607834522104175643, −4.49077073330353347212998581152, −3.557013077707001329171760927989, −2.52792909585195968136182090407, −2.21779976624956128182199867588, −0.91915313868821276331544199081, 0.46709360792189741248296805094, 1.17181363259096066595943064947, 2.43714693325708024097057289313, 4.051139579531600901748784543198, 4.62456989257631549592946414135, 5.302605328640997399333153883252, 6.19966685923673342984634144020, 7.16457905051989603044958001260, 7.92415052640587477285793915853, 8.44445427785525391196955548842, 9.52274132747865272496718070920, 10.062956735023295660493963931840, 10.87741611608391001470334849592, 12.29265373223744225584515079271, 12.81033789946333874731341128183, 13.81539734107585258002753686905, 14.09603179017906794865719123545, 15.14971697743097746220674417540, 15.9960587025007839659022060950, 16.48131750377176906217319861326, 17.424588878998611198740192938562, 17.669335709829127172821991053359, 18.54212193487321931691947977626, 19.61985009247095474277977535387, 20.24923923730630795121787815557

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