↑

Properties

Label 1-1400-1400.1371-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.535 - 0.844i$
Analytic cond. $6.50157$
Root an. cond. $6.50157$
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  + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + (−0.309 + 0.951i)33-s + (0.809 − 0.587i)37-s + (0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + (−0.309 + 0.951i)33-s + (0.809 − 0.587i)37-s + (0.809 + 0.587i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9071404611 - 0.4987046887i\)
\(L(\frac12)\) \(\approx\) \(0.9071404611 - 0.4987046887i\)
\(L(1)\) \(\approx\) \(0.8216616381 - 0.2312662885i\)
\(L(1)\) \(\approx\) \(0.8216616381 - 0.2312662885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
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

−20.948729666926832852485889599784, −20.21369332308309376748032368945, −19.712468387065000070170649596144, −18.46648605510485858530328383020, −17.79036724881787929271580274921, −17.1457389302226458307231811478, −16.289543629814706709653386695118, −15.59911003098776884096537164530, −14.99485166068530999492553762778, −14.2970290964404822582111444568, −13.16557067277446042862802301685, −12.48418356653335417879005421951, −11.55348417540374918719131908683, −10.75404728058149872118517139071, −10.18508337178883132495419279784, −9.32620902643994609930280986684, −8.67320983078017805427231199150, −7.50280863693565451231399450971, −6.79174392341023755467149890649, −5.593400345330398704771613956264, −4.88409075328811200274878701673, −4.40284109371333715016185699098, −2.96358108499259720456792475250, −2.56681590302061915207807974825, −0.73899575819955596012648480741, 0.61056740956899746136917546864, 1.90797709669236931579197834542, 2.530869191114314290290391530808, 3.76134725547781525524166777836, 4.84953330829454737140841800715, 5.85462426967245293191893238591, 6.30606486537137248405029369761, 7.54345589096970251884001789935, 7.85625980295339527549476797450, 8.87780708732201406752636306365, 9.8412021307903330398302758917, 10.91120038867488770561864620062, 11.36963762992997786274486109345, 12.44973218561883464886130188141, 12.87855794215752839100445546330, 13.72337176510388686355362882082, 14.45487254811424137001618358643, 15.31811023664407389614701039853, 16.3287907037756103441337413176, 17.06769141057157035698155234022, 17.56763884345004987243835089294, 18.580131433129388670825675690525, 19.143456597543563030585172927178, 19.5604429128492088911530309050, 20.78321890355574809141507581092

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