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Properties

Label 1-3311-3311.1545-r1-0-0
Degree $1$
Conductor $3311$
Sign $-0.739 + 0.672i$
Analytic cond. $355.816$
Root an. cond. $355.816$
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  + (0.251 + 0.967i)2-s + (−0.936 − 0.351i)3-s + (−0.873 + 0.486i)4-s + (0.646 − 0.762i)5-s + (0.104 − 0.994i)6-s + (−0.691 − 0.722i)8-s + (0.753 + 0.657i)9-s + (0.900 + 0.433i)10-s + (0.988 − 0.149i)12-s + (−0.525 − 0.850i)13-s + (−0.873 + 0.486i)15-s + (0.525 − 0.850i)16-s + (−0.525 + 0.850i)17-s + (−0.447 + 0.894i)18-s + (−0.420 + 0.907i)19-s + (−0.193 + 0.981i)20-s + ⋯
L(s)  = 1  + (0.251 + 0.967i)2-s + (−0.936 − 0.351i)3-s + (−0.873 + 0.486i)4-s + (0.646 − 0.762i)5-s + (0.104 − 0.994i)6-s + (−0.691 − 0.722i)8-s + (0.753 + 0.657i)9-s + (0.900 + 0.433i)10-s + (0.988 − 0.149i)12-s + (−0.525 − 0.850i)13-s + (−0.873 + 0.486i)15-s + (0.525 − 0.850i)16-s + (−0.525 + 0.850i)17-s + (−0.447 + 0.894i)18-s + (−0.420 + 0.907i)19-s + (−0.193 + 0.981i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.739 + 0.672i$
Analytic conductor: \(355.816\)
Root analytic conductor: \(355.816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3311,\ (1:\ ),\ -0.739 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3850310599 + 0.9956532412i\)
\(L(\frac12)\) \(\approx\) \(0.3850310599 + 0.9956532412i\)
\(L(1)\) \(\approx\) \(0.7970206203 + 0.2933749307i\)
\(L(1)\) \(\approx\) \(0.7970206203 + 0.2933749307i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.251 + 0.967i)T \)
3 \( 1 + (-0.936 - 0.351i)T \)
5 \( 1 + (0.646 - 0.762i)T \)
13 \( 1 + (-0.525 - 0.850i)T \)
17 \( 1 + (-0.525 + 0.850i)T \)
19 \( 1 + (-0.420 + 0.907i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (-0.163 + 0.986i)T \)
31 \( 1 + (0.963 + 0.266i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (0.550 - 0.834i)T \)
47 \( 1 + (-0.575 - 0.817i)T \)
53 \( 1 + (0.983 - 0.178i)T \)
59 \( 1 + (0.280 + 0.959i)T \)
61 \( 1 + (0.963 - 0.266i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.193 + 0.981i)T \)
73 \( 1 + (-0.791 + 0.611i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.712 - 0.701i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
97 \( 1 + (-0.753 - 0.657i)T \)
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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

−18.44808936357992815821364270277, −17.74218495965369992911191617303, −17.33027934462448268009222242106, −16.54071597225205920008233326097, −15.49512870377292954546637541127, −14.87346080770163534615106237618, −14.1838759289668308613847234709, −13.33210200940590206300229161739, −12.885380987191522866443316551098, −11.797158703127873338822829421, −11.3924185712649486391466630917, −10.85386990100159949647252057325, −10.08850138028793011318567159321, −9.43907928030169880596859298383, −9.038115504952483310479074706548, −7.56411024309278833132403371774, −6.59178473584751255046496344069, −6.20941872955925685227103184967, −5.129186889780264530616644751436, −4.65535695133422429572329104304, −3.88678398218285062347713571455, −2.72268555866738664991708907940, −2.29594337108954964675963726680, −1.123395482665587276674888859516, −0.24152204419627636584014699604, 0.75586041377494892189720807245, 1.5611874065209086919442565634, 2.77094211351376763357996436175, 4.0609879901155533172039683306, 4.68839421386969754820559857397, 5.57680187578766274257279064910, 5.73494864040870900321790566562, 6.72578800967567664255798972926, 7.29329337227761161253611814580, 8.34351855993384606857983431635, 8.67937581184484030343817683856, 9.96620233179187416894267237664, 10.2098968449480162695571314712, 11.39882313243632531700478415414, 12.30720221443844958746048205712, 12.883015976268786879874433638475, 13.1339044013782347764653128631, 14.07155299747698317386792288222, 14.91484281498567893221800212759, 15.57606500217087179685912271080, 16.428738411587615253218626287684, 16.86235650790589179871303655280, 17.49938780342366714293822808112, 17.82614227953180915164268071100, 18.70321234409399052800991092108

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