Properties

Label 1-1400-1400.237-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.728 + 0.684i$
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  + (0.587 + 0.809i)3-s + (−0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (0.951 − 0.309i)23-s + (−0.951 + 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.951 − 0.309i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s i·43-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s + (−0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (0.951 − 0.309i)23-s + (−0.951 + 0.309i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.951 − 0.309i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4512950510 + 1.139841185i\)
\(L(\frac12)\) \(\approx\) \(0.4512950510 + 1.139841185i\)
\(L(1)\) \(\approx\) \(0.9941848849 + 0.4309619501i\)
\(L(1)\) \(\approx\) \(0.9941848849 + 0.4309619501i\)

Euler product

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

−20.42546932171634907922494162104, −19.79895313739353956035406104556, −18.93357079435252645135477568828, −18.43135223593970722363423046492, −17.36734125047881231743386998405, −17.17365621474922722268016963276, −15.58424057549010180364064376309, −15.29946708217982911457227369691, −14.28306956951906216659046472055, −13.61079698767344899983241447825, −12.964417187739363544296335165442, −12.04316093080944915791872850775, −11.594968851853782982918721522848, −10.3341290152516663813027131123, −9.38235561925317433594364196884, −8.97824237005579637347618723055, −7.6820839487633080613757488614, −7.28172730798575533138469641815, −6.5976260055080311662440058047, −5.32629196402759861906978156617, −4.57937859511696867768333662327, −3.35685153472115292461044173130, −2.46379720688841748994860641602, −1.80981660654911136900887146237, −0.4127344794852048719308669054, 1.4056968604369863824533156970, 2.71299168936603289248923759786, 3.22766255568610635783824879575, 4.26698307979619721263234887434, 5.11459728097848156448372767128, 5.84805587467913535511991111196, 7.07437813440910907321916033591, 7.980469179491003368987067593786, 8.65043430007515975523890519139, 9.41882514690118605164617009630, 10.29903910939586715076145417295, 10.84500487068708928903862877430, 11.75120339616594842840692048665, 12.841297815378406294042559286103, 13.510510623429541818745276449637, 14.44563089033201496749181592767, 14.89762557438842907645185791004, 15.81167544082619940077836547685, 16.418480170314906187264984019066, 17.1557541521867351606323567051, 18.05339974016026334342767830108, 19.206963463236416191073011343, 19.43647825840798680324887660349, 20.484147404487459192992119477698, 20.99987166681654402979238587844

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