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Properties

Label 1-1400-1400.1237-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: $\chi_{1400} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01295459882 - 0.1552840162i\)
\(L(\frac12)\) \(\approx\) \(0.01295459882 - 0.1552840162i\)
\(L(1)\) \(\approx\) \(0.6416840845 - 0.04025704406i\)
\(L(1)\) \(\approx\) \(0.6416840845 - 0.04025704406i\)

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

−21.40329324870734811539337209701, −20.3929666619850929657628299187, −19.45376273822352482592655315630, −18.94516529258812787572460543410, −18.010979727002573314510465085360, −17.21590942692894841300059893171, −16.83537043112985191910435283627, −16.03617807270089410810450586913, −15.08945758587665132874080939628, −14.36779873966915962157901458611, −13.4277860423941274258202344458, −12.49574878623420677553725640793, −11.966830563198767550414735151123, −11.25898039544054264558616350530, −10.38628796542634921107881510796, −9.665856540297963623310640953526, −8.79423312773412905388574431996, −7.57025952694887460239538791526, −7.03824655541890458471948360088, −5.88178118727327940551804695047, −5.63183701208735023579363741049, −4.191369787545630600984606031325, −3.879601463080337722247879613054, −2.16391288643400952256062967187, −1.41398710114166084396838990260, 0.07135470704059578480914602077, 1.305192563849827105010184916956, 2.44230774842946805619278290747, 3.65624813292241503058758389068, 4.69617214803794429139597695944, 5.13505382950212637005609384781, 6.30620675473595452087315450730, 6.90218824660872087486637407301, 7.651005159530614483144500186124, 8.98488003560774763121622890340, 9.62220179415677509869732283457, 10.41844541052260423124084321124, 11.36399340661243989922216384695, 11.83951133962099081953673528658, 12.64586617538283670899306800901, 13.412968655621996420409901713864, 14.50846366820298632078354656802, 15.129697657270782458240385842046, 16.08917524038336358515081040347, 16.70262476093529827209941172490, 17.39235777753551809320486496474, 18.0799849028009951018219767018, 18.71507351713370375572317297955, 19.98334115292183760402993867859, 20.14243828189869388779514152016

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