Properties

Label 1-1400-1400.1083-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.937 - 0.347i$
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  + (0.743 − 0.669i)3-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (0.587 − 0.809i)13-s + (0.207 − 0.978i)17-s + (0.669 − 0.743i)19-s + (0.406 − 0.913i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.743 − 0.669i)33-s + (−0.994 − 0.104i)37-s + (−0.104 − 0.994i)39-s + (0.809 + 0.587i)41-s + i·43-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)3-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (0.587 − 0.809i)13-s + (0.207 − 0.978i)17-s + (0.669 − 0.743i)19-s + (0.406 − 0.913i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.743 − 0.669i)33-s + (−0.994 − 0.104i)37-s + (−0.104 − 0.994i)39-s + (0.809 + 0.587i)41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4963390879 - 2.767162640i\)
\(L(\frac12)\) \(\approx\) \(0.4963390879 - 2.767162640i\)
\(L(1)\) \(\approx\) \(1.212090469 - 0.7509973011i\)
\(L(1)\) \(\approx\) \(1.212090469 - 0.7509973011i\)

Euler product

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

−20.89105095720731447681015284675, −20.29312483648203733723077137317, −19.52416734799769183364889376876, −18.8353330083727267049000206199, −17.97494191099262901392354182592, −17.036412711272778363478253344696, −16.2809984425944226998274897949, −15.589858387135003093984628804891, −14.85083672977910617680813296326, −14.20955488821906048686875477614, −13.45888512110730603046517770567, −12.57218617231842131746646972265, −11.72288536132180407818726833043, −10.63007568350958174238837417225, −10.162308379020157496556761334560, −9.159102979718463639400680781983, −8.70825350150206281789361663628, −7.61438516302796713066814934805, −7.018681661803785342647669622789, −5.70824053007453629905242815738, −4.94793295447420713876345802842, −3.87589957011127854435827548855, −3.465906736532092307480349473755, −2.11219535905816104200472119573, −1.4764731572431569432989335493, 0.50524412591150126757516139558, 1.1029591195427602386646023869, 2.55198316863276232139327641189, 3.03263301017577873388073831444, 3.998641528359514550606380712838, 5.25931591337135636444101149756, 6.08796824623436662187863566893, 6.99689001224025600005557004565, 7.77015022600710521537407175033, 8.524975090465355085276948644, 9.17031488231747816468261470955, 10.10414129746454788567866573414, 11.17240854312734831438003548049, 11.781949554367418768587724531876, 12.9414196509366302828168729845, 13.28587767653056261242250189312, 14.14877552822143553914231375410, 14.750780920335016811876720037522, 15.81611364336861258787334842542, 16.25804967126161632822627997456, 17.567050476336949432375511301831, 18.05591020421739213312113242795, 18.88544568697461560146717736198, 19.37822015995219406685364231385, 20.4282672198029487752637100992

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