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Properties

Label 1-1400-1400.1003-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.588 - 0.808i$
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  + (0.207 − 0.978i)3-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (0.587 + 0.809i)13-s + (0.743 − 0.669i)17-s + (0.978 − 0.207i)19-s + (0.994 + 0.104i)23-s + (−0.587 + 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.669 − 0.743i)31-s + (−0.207 − 0.978i)33-s + (−0.406 + 0.913i)37-s + (0.913 − 0.406i)39-s + (−0.809 + 0.587i)41-s + i·43-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)3-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (0.587 + 0.809i)13-s + (0.743 − 0.669i)17-s + (0.978 − 0.207i)19-s + (0.994 + 0.104i)23-s + (−0.587 + 0.809i)27-s + (0.309 + 0.951i)29-s + (−0.669 − 0.743i)31-s + (−0.207 − 0.978i)33-s + (−0.406 + 0.913i)37-s + (0.913 − 0.406i)39-s + (−0.809 + 0.587i)41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.588 - 0.808i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.588 - 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.708764027 - 0.8700740923i\)
\(L(\frac12)\) \(\approx\) \(1.708764027 - 0.8700740923i\)
\(L(1)\) \(\approx\) \(1.224115823 - 0.3933266482i\)
\(L(1)\) \(\approx\) \(1.224115823 - 0.3933266482i\)

Euler product

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

−20.823542781716382559245828411625, −20.320235642692653850450865540604, −19.54353590630994935103625267425, −18.784583590053123298947053219088, −17.6467360813328873330400608282, −17.12884042635113095489471240955, −16.28933323762939609562538960548, −15.578941727990042979182604465, −14.87208769393479640561022484517, −14.24318994850721000987814213083, −13.41576499241511440799849665374, −12.36311466710712945037551750864, −11.636145011797350611002730501445, −10.68208507653643380020102010947, −10.13555083969819595504584451028, −9.25861939997222903748353303834, −8.60374765327554889107174908204, −7.727494989792043439464598870155, −6.69745237112529191508423548374, −5.59162528546325642185335256059, −5.08910302712655028105941258366, −3.725891471357078799526360115320, −3.56902510471604351934561584122, −2.26428099623391409056524295320, −1.00176106602458568603576770948, 0.99513074339170338962600389265, 1.573069654699874067356551672472, 2.92321862924207993476363032022, 3.501932824540965475203096045532, 4.80128336179770594354771949031, 5.795283467046986722388364025776, 6.61750943852839381299547417843, 7.23548807885732224639478542008, 8.111623116302008169229332161803, 9.06029550215077455656161239265, 9.472271508962245952458000946323, 10.93040032740620734324254979037, 11.60072785932870441868641735772, 12.15546234215888085060685136799, 13.131867311091657134563373883789, 13.88693909167101784282447150267, 14.29983442383618672742465472242, 15.243243066027613572446042989764, 16.40078582580131027672090102877, 16.8548627126110614618825610083, 17.80524064680671035181211070591, 18.644377808510827008615480812, 18.941849011681430599486220797346, 19.93993604446226533957430482066, 20.46664071412297636930253092435

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