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Properties

Label 1-1400-1400.1179-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: $\chi_{1400} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.347 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9448030819 - 1.357818614i\)
\(L(\frac12)\) \(\approx\) \(0.9448030819 - 1.357818614i\)
\(L(1)\) \(\approx\) \(1.145643489 - 0.4766794561i\)
\(L(1)\) \(\approx\) \(1.145643489 - 0.4766794561i\)

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

−21.04206608941831311104854372078, −20.36075193028240758061160812049, −19.59258999222255839346082629162, −18.82018834015388250911147136360, −18.2517861080631885061366873550, −16.88504403270814486301972665819, −16.51559543542700060924336993272, −15.67388349804084109501694057593, −15.00405558729790077902765107631, −14.18328912026730752178593083449, −13.504819598547736798613324872733, −12.883154252705860484207693644623, −11.49178554772765930589727797095, −10.974322848431402269332979740712, −10.26321604324328747237472382988, −9.06221730132862054722410416056, −8.83772560315380386818011678119, −7.9384951896444702817974816148, −6.88949127943633679524203815156, −5.918949498522784011703410127175, −5.02067669959865646790512927387, −4.04173359699520478846943268375, −3.41857510386352918302725726310, −2.452438598713238653875948012385, −1.375959993213142453049105140411, 0.56940432017611691848775972052, 1.88303289816829331068833751882, 2.46400057282091247570793140992, 3.59024924140591980635152244734, 4.4205941291703399661243262134, 5.581334305689212259380348105406, 6.602760340258083157963787286746, 7.1488174428755070227844092637, 8.06123436276907876448638368685, 8.83899090283019102020894691909, 9.43922447160037300311307649043, 10.6166828190202183554366436603, 11.28267015998934267559262016720, 12.44472081731189688729160069570, 12.953008503891764412340565961000, 13.5102898939432644489692985567, 14.52812801881542574210501424341, 15.20824614741558052095926259951, 15.71766503850340638549221421563, 17.04469256139005859847334714372, 17.71371750823176465310022480154, 18.26066872704973191405986752579, 19.13142225246588240170688283932, 19.82176423401452880916287773075, 20.46576427923372933333227984485

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