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Properties

Label 1-1400-1400.363-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.728 + 0.684i$
Analytic cond. $150.450$
Root an. cond. $150.450$
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

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9267807064 + 2.340780863i\)
\(L(\frac12)\) \(\approx\) \(0.9267807064 + 2.340780863i\)
\(L(1)\) \(\approx\) \(1.208011607 + 0.5893293894i\)
\(L(1)\) \(\approx\) \(1.208011607 + 0.5893293894i\)

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

−20.238860178136142242629377643788, −19.44827564104375622855900349002, −18.78668050974954143601313436088, −18.38560931669769588732817102279, −17.23209865177192210428610238834, −16.75284060000252034839024480283, −15.63855985427849129398994888977, −14.8412052421852364596493061793, −14.2129261964888491032076616494, −13.32357203217534067009194104135, −12.88274014790601463661034723121, −11.95903061220988371412372502867, −11.10809652510011772824744081136, −10.33581896741220003979016011870, −9.08928349460067271314860354364, −8.593089377303157465603262939155, −7.87235009423824909106545743766, −6.96558797781854967416434478587, −6.067330223238870863557146885194, −5.52287945050049650085649509936, −3.76456399386207548793794969114, −3.538170875516278355449458999851, −2.25443571149787619734974392939, −1.36624189692255818580829398124, −0.45671713408004438038481792124, 1.169915193465850648511386239286, 2.29905707901079452385178670681, 3.15255201618034901296353825032, 4.13777732992691738032026603560, 4.72662174767659699424158857903, 5.71586236584169363565328512745, 6.82464221673833130620792977464, 7.63384598471055567495812399287, 8.60026918402804329426156784800, 9.308648004505848723479472761698, 9.843081546528146362282642980751, 10.96108321907932995121559838754, 11.34693754589591261191318990331, 12.64313907179712457511930387724, 13.29341511697202362787201506062, 14.21050548315120223173361186784, 14.917495066254570002603238706855, 15.39383519577841007846389068913, 16.48663798190570387861759778639, 16.77473726668235292977674443282, 17.96116315845931132357889415900, 18.6917826776278439822696245250, 19.52080515522532714509202707610, 20.34616302984215859858439719999, 20.764493128812778013421917641989

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