↑

Properties

Label 1-1323-1323.1172-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.523 - 0.852i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.853 − 0.521i)2-s + (0.456 − 0.889i)4-s + (0.995 + 0.0995i)5-s + (−0.0747 − 0.997i)8-s + (0.900 − 0.433i)10-s + (0.853 − 0.521i)11-s + (0.661 + 0.749i)13-s + (−0.583 − 0.811i)16-s + (−0.222 + 0.974i)17-s − 19-s + (0.542 − 0.840i)20-s + (0.456 − 0.889i)22-s + (0.998 − 0.0498i)23-s + (0.980 + 0.198i)25-s + (0.955 + 0.294i)26-s + ⋯
L(s)  = 1  + (0.853 − 0.521i)2-s + (0.456 − 0.889i)4-s + (0.995 + 0.0995i)5-s + (−0.0747 − 0.997i)8-s + (0.900 − 0.433i)10-s + (0.853 − 0.521i)11-s + (0.661 + 0.749i)13-s + (−0.583 − 0.811i)16-s + (−0.222 + 0.974i)17-s − 19-s + (0.542 − 0.840i)20-s + (0.456 − 0.889i)22-s + (0.998 − 0.0498i)23-s + (0.980 + 0.198i)25-s + (0.955 + 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.523 - 0.852i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.523 - 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.014642410 - 1.686048194i\)
\(L(\frac12)\) \(\approx\) \(3.014642410 - 1.686048194i\)
\(L(1)\) \(\approx\) \(2.009675676 - 0.7433855806i\)
\(L(1)\) \(\approx\) \(2.009675676 - 0.7433855806i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.853 - 0.521i)T \)
5 \( 1 + (0.995 + 0.0995i)T \)
11 \( 1 + (0.853 - 0.521i)T \)
13 \( 1 + (0.661 + 0.749i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.998 - 0.0498i)T \)
29 \( 1 + (0.998 + 0.0498i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (-0.583 + 0.811i)T \)
43 \( 1 + (-0.411 + 0.911i)T \)
47 \( 1 + (-0.0249 - 0.999i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (0.270 + 0.962i)T \)
61 \( 1 + (-0.456 - 0.889i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.955 - 0.294i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.661 + 0.749i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.20202973145191670491630418590, −20.49382793488284085124063839915, −19.887327971269460717565882274736, −18.590788305912140485304854336481, −17.72466137918292432150874737351, −17.219008835158986641836580290970, −16.4933998301781164129791219522, −15.59155996740902144453860538214, −14.844704947616170774167044274306, −14.14988956340888746394548070720, −13.42047592224473463047797215916, −12.809198534119859008679161447242, −12.0391632510822951469549818332, −11.05342247103504682739489374011, −10.25973983957814588965165916178, −9.08503001830887069913422900310, −8.58621022570352233131328129895, −7.31367927179282327567254032403, −6.64512257160849974873454738837, −5.927366586298089671069122217690, −5.07594957406369757819256726046, −4.34148222522334665555051322615, −3.21858375586674583711293876003, −2.39541831976776563461726087805, −1.30253737335343986555401765418, 1.194981315024205927471793570606, 1.87708015304385667642887651723, 2.8699143109925260506306023765, 3.86366987259517460606966970105, 4.59928778675121072697736306620, 5.711719837650736734697453036815, 6.35929767672721850742837098627, 6.82931542704387834845419697348, 8.49815618395607830123283026304, 9.193482739845978677449794022408, 10.0672269113791779409828059569, 10.903295903171243996148632427730, 11.42468763066447861742101099794, 12.48143271548408468752381648088, 13.269204326648474401197291285582, 13.68245885818295796478956969448, 14.76824164686832039241226427116, 14.91494436508364306466188567505, 16.399232851970781147324724984241, 16.80732225635432999880444342286, 17.89235606934278218207410299905, 18.73932366015139551724471549262, 19.40168116521723342346886473529, 20.13045547462467702000377019013, 21.24626351607111772112537661825

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