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Properties

Label 1-1400-1400.3-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: $\chi_{1400} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.937 + 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01541062323 - 0.08591646701i\)
\(L(\frac12)\) \(\approx\) \(0.01541062323 - 0.08591646701i\)
\(L(1)\) \(\approx\) \(0.7181123807 - 0.1272608528i\)
\(L(1)\) \(\approx\) \(0.7181123807 - 0.1272608528i\)

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

−21.42617994045575314261627186754, −20.29797154552148251477661759370, −19.54360400444692704567626210603, −18.74018086071761386126849159587, −17.85291806752317490656283559191, −17.15437080608299048285052197147, −16.52097931180401560087803576363, −15.816244201711695405680871020788, −15.1104737694312124600096797460, −14.24436048219142422278964418491, −13.39187410915442963231334524431, −12.49545983621699269512601734109, −11.40100861778215881110717875165, −11.32898671534910175051563619029, −10.17406781511816129998100878207, −9.50540040482575558174841350893, −8.74206074103081092473693366876, −7.679153448851279236524489025379, −6.67642564439118733432643733455, −5.89717429696152358580510077958, −5.2013579931335382949405686591, −4.20307998467519681474239357332, −3.5450958779581861970955193102, −2.339500950172231216113123800765, −1.023315353478506560640438435928, 0.02303443989030909538876142104, 1.04191702638425119967521438034, 2.1056990275038062641028234416, 2.95851801853127743499690053729, 4.394252017437311378563969036125, 5.11848803910452053477217867099, 5.86222418931545833926072505311, 6.90249506432628636610373642311, 7.463015097837859116279575829856, 8.194355845756048090201841556852, 9.47726795148412035759534294614, 10.16205902608190189471452637976, 11.005633389681431296318593726455, 11.82169546784220286512447426962, 12.65137211861214604238136300495, 12.89942926275521467748122458508, 14.185178582550014906416382401726, 14.669866688596032234437570852297, 15.95257734421747410984577635081, 16.32902074666820698398402335052, 17.36969368563562082924380376266, 18.03218484239418554476426916047, 18.34556002220276452828044467005, 19.44862668345465129253902895671, 20.16178472863924013342008560458

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