↑

Properties

Label 1-1400-1400.333-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.588 + 0.808i$
Analytic cond. $150.450$
Root an. cond. $150.450$
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  + (0.207 + 0.978i)3-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (−0.587 + 0.809i)13-s + (−0.743 − 0.669i)17-s + (−0.978 − 0.207i)19-s + (0.994 − 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (0.669 − 0.743i)31-s + (0.207 − 0.978i)33-s + (0.406 + 0.913i)37-s + (−0.913 − 0.406i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)3-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (−0.587 + 0.809i)13-s + (−0.743 − 0.669i)17-s + (−0.978 − 0.207i)19-s + (0.994 − 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (0.669 − 0.743i)31-s + (0.207 − 0.978i)33-s + (0.406 + 0.913i)37-s + (−0.913 − 0.406i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.255874116 + 0.6394701170i\)
\(L(\frac12)\) \(\approx\) \(1.255874116 + 0.6394701170i\)
\(L(1)\) \(\approx\) \(0.8927300484 + 0.2703398240i\)
\(L(1)\) \(\approx\) \(0.8927300484 + 0.2703398240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.743 - 0.669i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.994 - 0.104i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.406 + 0.913i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (0.207 + 0.978i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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

−20.31079369681479736558311330430, −19.67767200423366959963439276956, −19.04996950181969686776295370943, −18.17917971981360139575665920178, −17.58854992253460730035594042528, −17.01172020069034467950024115965, −15.83619773524207487576469033803, −15.02517310382325469951526110732, −14.47049957348176732760950492840, −13.41061745574235985670494172901, −12.6890585428784114773549575585, −12.50682466010078084663561657732, −11.1762299867937252297149322010, −10.59513459255578051206111286737, −9.57168687199825240516650457992, −8.498555086335883037648355643183, −8.03545259526666619753044602995, −7.09471556450725163706943999323, −6.453415162161419554814676057809, −5.434376641296833195061398592204, −4.63112272333943892995937662494, −3.26804268509821980756704428833, −2.5225598552679485024927863138, −1.66192476098292739952559281403, −0.47698501018161921716239840728, 0.50352771011532019915161994560, 2.31562580558164401608242391581, 2.75380294751525996923408975440, 4.0359911112966954302941474834, 4.65739360815879613031580741351, 5.43634963877259008677013611269, 6.47023063641242810398629232730, 7.43151519250853937133247993178, 8.48714928184440294838093643100, 9.00346773912731724272700635965, 9.93377786867292954877305601702, 10.59308766981428839238722805952, 11.36512043659082727781109190488, 12.096385174289095570766988966972, 13.46799165856569135561615971975, 13.67473016323182690675397751823, 14.96489336147753065253449617722, 15.27952235187247440093527783890, 16.131599896361733679330145645163, 16.92323414060976952715713401676, 17.417389859670598475449460350818, 18.72086270521244001628960886135, 19.13672802323389328534377600679, 20.16855587102264723650695563196, 20.783971703768430462721710119198

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