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Properties

Label 1-1323-1323.34-r1-0-0
Degree $1$
Conductor $1323$
Sign $-0.477 + 0.878i$
Analytic cond. $142.176$
Root an. cond. $142.176$
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  + (−0.969 + 0.246i)2-s + (0.878 − 0.478i)4-s + (−0.542 − 0.840i)5-s + (−0.733 + 0.680i)8-s + (0.733 + 0.680i)10-s + (0.698 + 0.715i)11-s + (0.411 + 0.911i)13-s + (0.542 − 0.840i)16-s + (−0.365 − 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.878 − 0.478i)20-s + (−0.853 − 0.521i)22-s + (0.878 − 0.478i)23-s + (−0.411 + 0.911i)25-s + (−0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (−0.969 + 0.246i)2-s + (0.878 − 0.478i)4-s + (−0.542 − 0.840i)5-s + (−0.733 + 0.680i)8-s + (0.733 + 0.680i)10-s + (0.698 + 0.715i)11-s + (0.411 + 0.911i)13-s + (0.542 − 0.840i)16-s + (−0.365 − 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.878 − 0.478i)20-s + (−0.853 − 0.521i)22-s + (0.878 − 0.478i)23-s + (−0.411 + 0.911i)25-s + (−0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.477 + 0.878i$
Analytic conductor: \(142.176\)
Root analytic conductor: \(142.176\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (1:\ ),\ -0.477 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3783526339 + 0.6359965626i\)
\(L(\frac12)\) \(\approx\) \(0.3783526339 + 0.6359965626i\)
\(L(1)\) \(\approx\) \(0.6525224958 + 0.07744576043i\)
\(L(1)\) \(\approx\) \(0.6525224958 + 0.07744576043i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.969 + 0.246i)T \)
5 \( 1 + (-0.542 - 0.840i)T \)
11 \( 1 + (0.698 + 0.715i)T \)
13 \( 1 + (0.411 + 0.911i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.878 - 0.478i)T \)
29 \( 1 + (-0.0249 - 0.999i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.456 + 0.889i)T \)
43 \( 1 + (0.542 - 0.840i)T \)
47 \( 1 + (-0.270 + 0.962i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (0.124 + 0.992i)T \)
61 \( 1 + (-0.878 - 0.478i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.988 + 0.149i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.411 - 0.911i)T \)
89 \( 1 + (-0.826 - 0.563i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−20.14343093232139122831446971613, −19.62822141853326849978946536636, −19.05648590999955474708838231459, −18.21563083901459704878335564028, −17.65120023873955320168082247323, −16.81949795142195822695511130899, −15.96449400468002901282239698682, −15.26629688119167921097467182294, −14.67100886795535445142270213903, −13.46701644646900549975726112998, −12.63258086217608965668603271068, −11.611207539901447067794148829653, −11.03044517636755559004645430183, −10.5848388577541905642075427557, −9.48972840265015700252582510699, −8.71515455024682099572685986188, −7.96998412900836743895722836958, −7.13522345291146484767480995537, −6.46197673177019683684911773278, −5.52583960759933003187102977787, −3.88040580171155066030621120226, −3.32819200884586864693141333645, −2.43269182696882800082331872979, −1.18183522650454243018384877545, −0.24349180356801704927936114158, 1.01374714219258135609266416203, 1.64936373265482220305275537843, 2.9216929476153951248713027956, 4.1857674363337083250275482687, 4.94981839683836080254404988405, 6.08314787280669446673923964928, 6.956759723974515446833879784923, 7.62310472230588892963586262594, 8.58652379837194114304457411560, 9.170821067181896133263456888069, 9.790782835948376616629358901511, 10.87986188601413277047158285703, 11.8628232258078955362311290560, 12.020875329432225460155832949643, 13.31086398471521279045790598066, 14.29434543685664817520429576862, 15.10853066509581400253433064926, 15.90647971061088461315194092981, 16.516650275550719806132608450453, 17.07820310684941937171994474883, 17.937828988446940688959356286571, 18.79625999078572551740899702676, 19.34303207042673528412102334213, 20.37735682866758575115194660803, 20.499533769310176315626900174468

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