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Properties

Label 1-1323-1323.1238-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.878 - 0.477i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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

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

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.878 - 0.477i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.878 - 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.166428370 - 0.8042704111i\)
\(L(\frac12)\) \(\approx\) \(3.166428370 - 0.8042704111i\)
\(L(1)\) \(\approx\) \(2.081553727 - 0.1190912124i\)
\(L(1)\) \(\approx\) \(2.081553727 - 0.1190912124i\)

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

−21.200369349515709609151762131485, −20.51259657679291281029858223011, −19.465999524145140535706269987963, −18.79084368203530977262708573017, −18.23419305391461553349396154192, −17.0597655217290697724660837096, −16.19953662576989193702294491441, −15.591882965498521977550656459552, −14.553752280630351623666318119424, −14.07363598477321477332839026163, −13.51560752084143753253745733897, −12.55881952868081975806518520081, −11.776165616518130156237867189154, −10.86109056283520295469785620472, −10.4300358057283182170549573090, −9.55378563399444233163597959027, −8.29434773229374307043804105906, −7.327861604766992505255386032, −6.44752866170684349959384814400, −5.82545418215887787007881957192, −5.10068664381276817656676516451, −3.7171178667476443015145321651, −3.35790730817788354416519896408, −2.17337296546048901842942884607, −1.46880880616482727844287644128, 0.92140095161315188249500490513, 2.26345010095398196891863969461, 2.859966881712822788858233491455, 4.21837349142181486231537905610, 4.81695899864529847235245310465, 5.66134941694484266307609020412, 6.224889060153064151222251446813, 7.60571771840716580483050241637, 7.88190304614267933811703269513, 9.18974280509479513165553554502, 9.96897705223050929741413174190, 10.93423608010720542317923887561, 11.8417920921029546790960693692, 12.64922126316512855579178849036, 13.16578184178224969795807904424, 13.82662468960112178972972071153, 14.67701451516233800875775094710, 15.73592595971587279760325124764, 15.95290586788370927814075919937, 16.97832575083158181102490500124, 17.717326626734181354796862897792, 18.37775921086651366825022048887, 19.89365715912325142324692830333, 20.2592853403234953205846345173, 20.963400879535444097439179652441

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