Properties

Label 1-1400-1400.429-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.808 + 0.588i$
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  + (−0.978 − 0.207i)3-s + (0.913 + 0.406i)9-s + (−0.913 + 0.406i)11-s + (−0.809 + 0.587i)13-s + (−0.669 − 0.743i)17-s + (0.978 − 0.207i)19-s + (0.104 − 0.994i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.669 + 0.743i)31-s + (0.978 − 0.207i)33-s + (0.913 + 0.406i)37-s + (0.913 − 0.406i)39-s + (−0.809 + 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)3-s + (0.913 + 0.406i)9-s + (−0.913 + 0.406i)11-s + (−0.809 + 0.587i)13-s + (−0.669 − 0.743i)17-s + (0.978 − 0.207i)19-s + (0.104 − 0.994i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.669 + 0.743i)31-s + (0.978 − 0.207i)33-s + (0.913 + 0.406i)37-s + (0.913 − 0.406i)39-s + (−0.809 + 0.587i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.808 + 0.588i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.808 + 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7411429044 + 0.2410345533i\)
\(L(\frac12)\) \(\approx\) \(0.7411429044 + 0.2410345533i\)
\(L(1)\) \(\approx\) \(0.7098049261 + 0.009461518298i\)
\(L(1)\) \(\approx\) \(0.7098049261 + 0.009461518298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.978 - 0.207i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−20.83998337355958459060116303579, −20.01120561128226921250081821438, −19.11834774389150194387469204698, −18.273713488111435094732959660649, −17.681052823854174304850453207230, −17.01145235944855904390525725086, −16.17821638764721465216879132393, −15.52468004829005344586865092715, −14.88991549497251014879690884115, −13.68251732651626914653128978525, −12.94872741571293687978616910324, −12.28076958868637347949246790956, −11.37518389870512468639311501181, −10.75925519039891465198305729955, −9.998020893603505226037177090980, −9.27686558514528446653072002774, −8.01390706878527227121128307190, −7.38558312755774722904852860503, −6.38091654898915907360438104508, −5.48838987200466661515150437712, −5.04309983749376719707720876877, −3.93844106791141248212963743292, −2.971910863211219171066356784451, −1.75152578700491607133202630908, −0.48475389646780562866526905814, 0.77959135548514412207295310700, 2.07510682289208804875112272010, 2.89629649645658705431844195926, 4.55207442417118401265792348012, 4.77075916126049788971469332124, 5.81467752524435359835136752500, 6.74713493681893303200630923777, 7.34481894214959231793535647219, 8.204158560551899949737493686565, 9.52433866851035686332623180208, 9.99823776525758456555375609130, 11.00395915021208608080872287408, 11.62802869652293218715037189307, 12.36861353472433510905892278285, 13.11571884089070804180741104667, 13.867064185976488507844381374174, 14.88639691966108947967699570316, 15.83987770441563284805747774867, 16.24839646704303943124285225264, 17.236784820090961883619066741700, 17.79778277351855298113994769862, 18.519602476904231830237410585833, 19.15185662612748066691415334512, 20.215726034792091078187458589769, 20.889597916136818325127527433354

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