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Properties

Label 1-1400-1400.1277-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.986 - 0.165i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.663876735 - 0.2222344273i\)
\(L(\frac12)\) \(\approx\) \(2.663876735 - 0.2222344273i\)
\(L(1)\) \(\approx\) \(1.685602400 - 0.03654623256i\)
\(L(1)\) \(\approx\) \(1.685602400 - 0.03654623256i\)

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

−20.63087206121715832525030482726, −20.10086005625732373931559493581, −19.28794339814858514386146369743, −18.81868456913431469989570313550, −17.84236800591937593311051974533, −17.10529362145939997641605385078, −16.11648127643222654422206663246, −15.428465389887553044126070475553, −14.57460222914114019421998175808, −14.1469558515881311500271070009, −13.089345440438359636940564300356, −12.71980639623011585042745498466, −11.50692201074217125904718168158, −10.832859866745225569948846669448, −9.71218960018316778269435766818, −9.04200678447025399946235138289, −8.531847358929966376156523867823, −7.43484499783159552751987726455, −6.83772422285361765685510475921, −5.86700200343383516689114792105, −4.66770684858642741856965803175, −3.71905239642704835943159928675, −3.22458158587886872671664579578, −1.87012020528986605213971412433, −1.28055716643492691745249109114, 1.07726049819062714865464101381, 1.95689085601772889669812595548, 3.17646034078592118281009077484, 3.67438027529642714926709932660, 4.636378895988486117288428843205, 5.75982951891236853454303880391, 6.69228050163409208216656900160, 7.57135524636550407787861742353, 8.353241560462441829802782325175, 9.09025107066953582103610561957, 9.73526447618163458622469739306, 10.6556307504702485713017894059, 11.54994183170628393870195727246, 12.44354710797320993193911943979, 13.33337517451682180784475372450, 13.928799816595304710646190830144, 14.70219505030806840618420723206, 15.26543154631255557312768627301, 16.329391063713794494435120713888, 16.71724262991311147948907214560, 17.98359445005884762013842120240, 18.72268855144603406258324317869, 19.16369006017635455918509076754, 20.2579313532818070115704350010, 20.6126264357268635757200287052

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