↑

Properties

Label 1-803-803.746-r0-0-0
Degree $1$
Conductor $803$
Sign $0.642 - 0.766i$
Analytic cond. $3.72911$
Root an. cond. $3.72911$
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  + (0.559 + 0.829i)2-s + (−0.978 + 0.207i)3-s + (−0.374 + 0.927i)4-s + (0.559 − 0.829i)5-s + (−0.719 − 0.694i)6-s + (0.669 − 0.743i)7-s + (−0.978 + 0.207i)8-s + (0.913 − 0.406i)9-s + 10-s + (0.173 − 0.984i)12-s + (0.438 − 0.898i)13-s + (0.990 + 0.139i)14-s + (−0.374 + 0.927i)15-s + (−0.719 − 0.694i)16-s + (−0.104 − 0.994i)17-s + (0.848 + 0.529i)18-s + ⋯
L(s)  = 1  + (0.559 + 0.829i)2-s + (−0.978 + 0.207i)3-s + (−0.374 + 0.927i)4-s + (0.559 − 0.829i)5-s + (−0.719 − 0.694i)6-s + (0.669 − 0.743i)7-s + (−0.978 + 0.207i)8-s + (0.913 − 0.406i)9-s + 10-s + (0.173 − 0.984i)12-s + (0.438 − 0.898i)13-s + (0.990 + 0.139i)14-s + (−0.374 + 0.927i)15-s + (−0.719 − 0.694i)16-s + (−0.104 − 0.994i)17-s + (0.848 + 0.529i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 803 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 803 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(803\)    =    \(11 \cdot 73\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(3.72911\)
Root analytic conductor: \(3.72911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{803} (746, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 803,\ (0:\ ),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001496395 - 0.4670593558i\)
\(L(\frac12)\) \(\approx\) \(1.001496395 - 0.4670593558i\)
\(L(1)\) \(\approx\) \(1.013237561 + 0.1395189248i\)
\(L(1)\) \(\approx\) \(1.013237561 + 0.1395189248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
73 \( 1 \)
good2 \( 1 + (0.559 + 0.829i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (0.559 - 0.829i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.438 - 0.898i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.615 + 0.788i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.0348 - 0.999i)T \)
31 \( 1 + (-0.997 + 0.0697i)T \)
37 \( 1 + (-0.615 - 0.788i)T \)
41 \( 1 + (-0.882 - 0.469i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.882 - 0.469i)T \)
53 \( 1 + (0.559 + 0.829i)T \)
59 \( 1 + (-0.882 + 0.469i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.961 - 0.275i)T \)
79 \( 1 + (0.438 - 0.898i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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

−22.145947029260143952273033500744, −21.54121064494112455748561612536, −21.36081915813401204245245859224, −19.97987689067323118632403708046, −18.9496714248310643153307698265, −18.40485393328509075366436713451, −17.87751445973134533345473753293, −16.9593518158323254297193281335, −15.72096684977860496892389049609, −14.873591948049749144710058249139, −14.18411839553766528725992841264, −13.213476246406138869899411507745, −12.48917230255019061112449692806, −11.51546882546484416035246987628, −11.08493144070178342134165924139, −10.34773402672274399662458469635, −9.408929404143168249527519835065, −8.31751812777705274270344772421, −6.689297379550193062382021292071, −6.28403740850883750864310085832, −5.323601496433988299360814664551, −4.54637641592828172630799413054, −3.398205867561505430092473957765, −1.989280902693133824459994305550, −1.6736036241052827140932909258, 0.45970477869891572209044332352, 1.85780120964754116441857282420, 3.70788212092793651178020230838, 4.410542439975954489034965195201, 5.36843661878959897960708049789, 5.74636799066276646517772540893, 6.85978924766023222180130137623, 7.76649372415434211058933719084, 8.62907267186028583224429167589, 9.76710555498325909225866228431, 10.605715270168563069956301468570, 11.73165905181282055368741492272, 12.38713439660458724003461637002, 13.33085369628172365332510119240, 13.85432589534657081053784940432, 14.97429543654136804751858083486, 15.870111731923498312422146043779, 16.53837902534760506363252251164, 17.15987352220980220862427893791, 17.81001447793834224914875865453, 18.384019438558544885286239961075, 20.18190856839997864762286946340, 20.81828804371765324647728676759, 21.42150999679440368975150838733, 22.31854533983766205808165771762

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