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Properties

Label 1-1400-1400.187-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.986 + 0.165i$
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  + (0.994 − 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

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.986 + 0.165i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.804439441 + 0.2339608980i\)
\(L(\frac12)\) \(\approx\) \(2.804439441 + 0.2339608980i\)
\(L(1)\) \(\approx\) \(1.411505217 + 0.02910691424i\)
\(L(1)\) \(\approx\) \(1.411505217 + 0.02910691424i\)

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

−20.62860452738975248769526290596, −19.69136313456868020940584738952, −19.31657333265775334765055124266, −18.45863926522219688718825134524, −17.62354220396379428180210290316, −16.796932938604703047927563497348, −15.819806094318041143043157833465, −15.15581351172768179601890174774, −14.663575237237090082884725286963, −13.674428847414240368911905449539, −13.01492165290157873105988013552, −12.431301073887938167285309726205, −11.20453434683123734269148265594, −10.39062080888295459840733316130, −9.689542609526768345831935027026, −8.84201212661186474940906857533, −8.15445178395271424460712298223, −7.29999141753797209615748015176, −6.638099733676297178650663470765, −5.25313404475067851582193244146, −4.5506075476503172633367523904, −3.64019193797445315939410131879, −2.48825626636947115429339000958, −2.14849591821887072618240285798, −0.60682173423604797276391531671, 0.744917596486339099771828919, 1.99098764166410230634069204074, 2.77676320148571934076012340862, 3.49437178277001920232845593748, 4.66679202391027907769046512338, 5.36356320365212586567186681572, 6.62559697857565693101966269579, 7.51078240443102970235536874303, 7.997112353200513043531189696875, 8.92902412528496985115721604771, 9.76420118290420264074048343336, 10.32195852278693730401977462837, 11.4294374407984370984730760349, 12.348315059573733429949099369323, 13.1789629398865093271712777220, 13.64963757530306361678180717294, 14.644362480474134224538546341917, 15.173750352899025393629378847, 15.93959448611769771846604907456, 16.77682090457988825318888049402, 17.77329410859055350058402766812, 18.5709285381266200500795598016, 19.06826013708508247737941730592, 19.95826477713000887196879034563, 20.611944336840099832852791090900

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