Properties

Label 1-1400-1400.709-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.637 - 0.770i$
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  + (−0.104 − 0.994i)3-s + (−0.978 + 0.207i)9-s + (0.978 + 0.207i)11-s + (0.309 − 0.951i)13-s + (−0.913 + 0.406i)17-s + (0.104 − 0.994i)19-s + (−0.669 + 0.743i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.913 − 0.406i)31-s + (0.104 − 0.994i)33-s + (−0.978 + 0.207i)37-s + (−0.978 − 0.207i)39-s + (0.309 − 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)3-s + (−0.978 + 0.207i)9-s + (0.978 + 0.207i)11-s + (0.309 − 0.951i)13-s + (−0.913 + 0.406i)17-s + (0.104 − 0.994i)19-s + (−0.669 + 0.743i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.913 − 0.406i)31-s + (0.104 − 0.994i)33-s + (−0.978 + 0.207i)37-s + (−0.978 − 0.207i)39-s + (0.309 − 0.951i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5214826464 - 1.107426003i\)
\(L(\frac12)\) \(\approx\) \(0.5214826464 - 1.107426003i\)
\(L(1)\) \(\approx\) \(0.8879245623 - 0.4442799189i\)
\(L(1)\) \(\approx\) \(0.8879245623 - 0.4442799189i\)

Euler product

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

−21.22841747173063998682718961187, −20.32993849472566578975224414444, −19.73495276413299900385868224708, −18.88592033809700697666869494322, −17.91032180610623395901122446128, −17.18736832424878442185336243090, −16.30857304360605534096272136896, −16.02109754754212377277112353499, −14.99848243522777522853788276965, −14.12227834858147474617377735664, −13.8880729773463697597017845766, −12.366140887925793149816747257853, −11.80051920254417726433490470995, −10.983369378692389076971344639661, −10.28167598333751308814811818729, −9.32534351517158222967304866692, −8.8655237134308508699118735524, −7.96778660291214471927441052349, −6.543879226842324686912819593618, −6.22239912936465852261600385279, −4.93273291698712959045579618050, −4.279646841701436735257461298431, −3.55611374409125361087804113702, −2.48931298481154537332520006308, −1.253816954845625172053530084988, 0.50611043859559685220079663813, 1.58007377605179876175109466944, 2.47696711454119533904445488627, 3.488341726700517375913922590364, 4.56504196867751905079027778943, 5.63841156302808511081282218463, 6.413503373651462023106818884026, 7.03550093556158938818987279960, 8.01291111044910751664505208100, 8.64855764128633932568171669939, 9.56019706709563470019897318370, 10.63757390386000044442911367366, 11.449268831438285479080907268246, 12.0786468868865114577329670595, 12.91166313366769762517034834495, 13.59635532489805703582410083641, 14.21251488829585938198227918442, 15.28499092505502429072593512603, 15.840597100956279619011826973958, 17.20199023825040856150290845329, 17.48501808646215379124285316586, 18.094780151482171129832959220278, 19.20355132757811010608662487009, 19.653410259631523138204428457412, 20.24591568249672738697937203609

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