Properties

Label 1-1400-1400.163-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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.994 − 0.104i)3-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.951 − 0.309i)13-s + (−0.406 − 0.913i)17-s + (0.104 − 0.994i)19-s + (−0.743 − 0.669i)23-s + (0.951 − 0.309i)27-s + (−0.809 + 0.587i)29-s + (−0.913 + 0.406i)31-s + (−0.994 − 0.104i)33-s + (−0.207 − 0.978i)37-s + (−0.978 − 0.207i)39-s + (0.309 − 0.951i)41-s + i·43-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)3-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.951 − 0.309i)13-s + (−0.406 − 0.913i)17-s + (0.104 − 0.994i)19-s + (−0.743 − 0.669i)23-s + (0.951 − 0.309i)27-s + (−0.809 + 0.587i)29-s + (−0.913 + 0.406i)31-s + (−0.994 − 0.104i)33-s + (−0.207 − 0.978i)37-s + (−0.978 − 0.207i)39-s + (0.309 − 0.951i)41-s + i·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: $\chi_{1400} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4915669005 - 1.043896612i\)
\(L(\frac12)\) \(\approx\) \(0.4915669005 - 1.043896612i\)
\(L(1)\) \(\approx\) \(1.104207682 - 0.2868901200i\)
\(L(1)\) \(\approx\) \(1.104207682 - 0.2868901200i\)

Euler product

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

−21.00675472639682783403388250041, −20.25043476890884794042907086583, −19.68906222039843513512349242351, −18.80748020176622999135656525155, −18.32386180719860041617195279462, −17.24582551439020155074191222253, −16.47763064949599700752165669823, −15.52709341477978366462102716925, −15.020357491290448883487150721129, −14.28637032829956255831933123300, −13.441582981634000500273877711631, −12.81205505262231319903180835016, −12.01890207669125221984604811660, −10.888737337116148302291311516463, −9.98455954888174447788297398462, −9.58875154275394410434726587633, −8.46398526418442692703108947573, −7.835733845455575854791969416452, −7.21265731745453274796919183474, −6.04239456018199742554702751534, −5.0482717509707049227263941837, −4.120402012827152399158756208152, −3.35513667685768187882510266462, −2.25729918814022596674499857074, −1.706639109365546820957107454751, 0.33705842623251895775984344237, 1.9166010756174796837436422067, 2.6438628220643179565328010227, 3.3603641105017094403966286161, 4.57263702737919646524127935590, 5.19802448833272715199602667209, 6.476496485251452696852707261602, 7.49401985571211997254631370916, 7.777837131679175326802898003241, 9.005761755870804063604225557660, 9.39733084960814491161604752662, 10.43892304523378387389410093973, 11.12021810601889201359620615037, 12.42281779892821571675585491813, 12.8417243309047788514444808818, 13.776220814397244384209666778615, 14.34489575638081273327542747918, 15.2044053852792584218068581820, 15.848543945013770212594290482234, 16.54983432508693963104322431346, 17.90324352681230165770382422094, 18.14789456412112328136554232784, 19.16997392803103455153755483296, 19.82531672187019774208173290278, 20.43433781441016649925740988041

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