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Properties

Label 1-1400-1400.171-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.347 - 0.937i$
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  + (−0.669 + 0.743i)3-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)17-s + (−0.669 − 0.743i)19-s + (−0.913 + 0.406i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.978 + 0.207i)31-s + (−0.669 − 0.743i)33-s + (0.104 + 0.994i)37-s + (0.104 − 0.994i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)3-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)17-s + (−0.669 − 0.743i)19-s + (−0.913 + 0.406i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.978 + 0.207i)31-s + (−0.669 − 0.743i)33-s + (0.104 + 0.994i)37-s + (0.104 − 0.994i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.347 - 0.937i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.347 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08861197595 - 0.1273482197i\)
\(L(\frac12)\) \(\approx\) \(0.08861197595 - 0.1273482197i\)
\(L(1)\) \(\approx\) \(0.6452904191 + 0.1565699485i\)
\(L(1)\) \(\approx\) \(0.6452904191 + 0.1565699485i\)

Euler product

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

−21.18680863575110265705152943706, −20.10480271002414512294094288070, −19.3508975389805828943843850193, −18.76059919627217481387064475823, −18.02904712297956481654439344665, −17.31432746971615420670578863775, −16.39955425993101669624263673496, −16.18080985211062862487116160490, −14.608612968398671140962945159018, −14.33651507412801273891204773247, −13.13200998755381540399870978198, −12.60224734201810994578027802284, −11.96005834561413965347955307201, −10.95424740133198895976604961916, −10.47356027740437567528512531876, −9.44018530388924393188145624240, −8.21298511815584045959519102635, −7.792227933409738599379209959163, −6.84675794658486081986949900236, −5.77853937155781898205481140872, −5.55855278542337377253586945707, −4.29191336059767812017125445427, −3.18682557444106837370802055981, −2.173379432862509395643839651064, −1.14105187312548180677750752315, 0.06892898293898049890779482156, 1.64754615252225872210739253211, 2.73166411893236640591323277358, 3.92501573112017928227478585800, 4.58378091552321352067062664384, 5.344509662042581754163130121721, 6.25197263031724794538319085241, 7.13128850439899089934208511993, 7.9471098627040632023322709690, 9.321726078080320564557325302032, 9.62013696880487448849635452112, 10.4810918709576311655664920335, 11.304723333354310883839866163410, 12.13263473801616015957530017386, 12.58907305224685239454498418100, 13.81698060973269135342107163789, 14.71216883709565836137844845803, 15.23513415329602392309653277487, 16.0815668875796637829986092177, 16.82171639736662224977873794776, 17.444106099023190124264718720184, 18.08968343003477477705692150973, 19.09521321620833101513471668812, 19.88924201718755409417742983128, 20.75650674376827550291792130459

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