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Properties

Label 1-1400-1400.1139-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.103 - 0.994i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 − 0.406i)3-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.309 − 0.951i)13-s + (−0.104 − 0.994i)17-s + (−0.913 − 0.406i)19-s + (−0.978 + 0.207i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.104 − 0.994i)31-s + (0.913 + 0.406i)33-s + (0.669 − 0.743i)37-s + (−0.669 − 0.743i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.309 − 0.951i)13-s + (−0.104 − 0.994i)17-s + (−0.913 − 0.406i)19-s + (−0.978 + 0.207i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.104 − 0.994i)31-s + (0.913 + 0.406i)33-s + (0.669 − 0.743i)37-s + (−0.669 − 0.743i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.103 - 0.994i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.103 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509860397 - 1.360970619i\)
\(L(\frac12)\) \(\approx\) \(1.509860397 - 1.360970619i\)
\(L(1)\) \(\approx\) \(1.351636140 - 0.4213567804i\)
\(L(1)\) \(\approx\) \(1.351636140 - 0.4213567804i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−21.17774638624396185948494711840, −20.029921856783186634497419068370, −19.538589617275885740789227174404, −18.97411216768928615272684514152, −18.10483766849174813375404727352, −16.86209875137865635068827008087, −16.559402257292478201577268290510, −15.55019191561573675053635565117, −14.80410683117151704427508586153, −14.16513080717093648718253559825, −13.585798459191677274007441510200, −12.60173534128297494771806860155, −11.77567784196813706063276595031, −10.79371011772319994791469668855, −10.05740417116073828019031415128, −9.27036581335676344754575896642, −8.43184450429826178382593951953, −8.019772414345999439756837930549, −6.70209298636780677868799131001, −6.14123600304007809149272110650, −4.734516319099844755364524306251, −4.09150860904902629749361809716, −3.31778067388049008158718412152, −2.23746660879267533387423840211, −1.43009053458829039595135999214, 0.67902525066661075066936686327, 1.97495581627369188291540247509, 2.597119028615067345865747314245, 3.68211970529408883972090939176, 4.44276337550879270419161401957, 5.54930903651754167948787593194, 6.70744638928719591082139613540, 7.24421346059310695845283678397, 8.13763230181720510645358137324, 8.866612910673226217397056556599, 9.71519832558247139215789604702, 10.31440657204710329342804006557, 11.58690698539050390602199976425, 12.2977406958326591076300647693, 13.02810946904651913490089830778, 13.75463539154719149022782755491, 14.59017788396185642283915996669, 15.13565499739071419721887709109, 15.87584629525008138165628882930, 16.94872321271127293494674803343, 17.82962812023847701554384323883, 18.27926366693234914534854242856, 19.28348127071546993981538610971, 20.05217071877533694977973057894, 20.23939695999420284104231678728

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