↑

Properties

Label 1-4312-4312.1565-r0-0-0
Degree $1$
Conductor $4312$
Sign $0.460 - 0.887i$
Analytic cond. $20.0248$
Root an. cond. $20.0248$
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  + (−0.887 + 0.460i)3-s + (0.772 − 0.635i)5-s + (0.575 − 0.817i)9-s + (−0.858 − 0.512i)13-s + (−0.393 + 0.919i)15-s + (−0.280 + 0.959i)17-s + (0.104 + 0.994i)19-s + (0.826 − 0.563i)23-s + (0.193 − 0.981i)25-s + (−0.134 + 0.990i)27-s + (−0.983 − 0.178i)29-s + (−0.978 + 0.207i)31-s + (0.646 − 0.762i)37-s + (0.998 + 0.0598i)39-s + (−0.0448 + 0.998i)41-s + ⋯
L(s)  = 1  + (−0.887 + 0.460i)3-s + (0.772 − 0.635i)5-s + (0.575 − 0.817i)9-s + (−0.858 − 0.512i)13-s + (−0.393 + 0.919i)15-s + (−0.280 + 0.959i)17-s + (0.104 + 0.994i)19-s + (0.826 − 0.563i)23-s + (0.193 − 0.981i)25-s + (−0.134 + 0.990i)27-s + (−0.983 − 0.178i)29-s + (−0.978 + 0.207i)31-s + (0.646 − 0.762i)37-s + (0.998 + 0.0598i)39-s + (−0.0448 + 0.998i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4312\)    =    \(2^{3} \cdot 7^{2} \cdot 11\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(20.0248\)
Root analytic conductor: \(20.0248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4312} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4312,\ (0:\ ),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9353263425 - 0.5685379263i\)
\(L(\frac12)\) \(\approx\) \(0.9353263425 - 0.5685379263i\)
\(L(1)\) \(\approx\) \(0.8552230306 - 0.04810204658i\)
\(L(1)\) \(\approx\) \(0.8552230306 - 0.04810204658i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.887 + 0.460i)T \)
5 \( 1 + (0.772 - 0.635i)T \)
13 \( 1 + (-0.858 - 0.512i)T \)
17 \( 1 + (-0.280 + 0.959i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (-0.983 - 0.178i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.646 - 0.762i)T \)
41 \( 1 + (-0.0448 + 0.998i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.992 + 0.119i)T \)
53 \( 1 + (-0.971 - 0.237i)T \)
59 \( 1 + (0.842 - 0.538i)T \)
61 \( 1 + (-0.971 + 0.237i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.134 - 0.990i)T \)
73 \( 1 + (0.992 - 0.119i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.858 + 0.512i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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

−18.56384242274590513141310342943, −17.694395846105264925027382870081, −17.18625466406502612066199435796, −16.70581238905408657983136084239, −15.816869323919435447247619520103, −15.061280843061065983551354277122, −14.33514370668951113524289302594, −13.52022819213515745186135346992, −13.16142561345915855047617624226, −12.30716995366133514434867060669, −11.43026459690800972808193469334, −11.13989791845643672907539956440, −10.34312680519661682637834470273, −9.446431357416461424396417899124, −9.15420732980671679237859243497, −7.69333126918390506567871093000, −7.143078577740975202682148133293, −6.71889451977472167347242313496, −5.85713987672645350001197943339, −5.1441156339813018732872711855, −4.65383835161776563513030598995, −3.38881083121798152888508456270, −2.45341396752907687476100397764, −1.90547966349704880080912491632, −0.846241443226024474184652779217, 0.41524500272316748110057629757, 1.4530281317451761147283686, 2.20415770205519700675489453840, 3.39713006966366485348821889153, 4.20874624844312491506500506706, 4.96305790068348668320498961058, 5.57146593415342268429065581259, 6.08801564304555133865027083848, 6.9236225248466356470100875042, 7.81228997175310003576491150889, 8.71025764963242169236473853358, 9.45882681516168070235415088568, 9.97858102746441308389161770717, 10.66675185452175543309730805204, 11.25544104967935313880049782093, 12.338924157114923536257288214046, 12.62596096859647539718493436621, 13.21116550657379114131606666854, 14.331128588396382917454832959055, 14.87858385069389210212471668745, 15.55715371792985423421742739734, 16.59070198202374393390882122194, 16.7681204032801772908557707194, 17.370567613834649176698809662342, 18.09043390406118218842684628581

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