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Properties

Label 1-1400-1400.1397-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.481 + 0.876i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.481 + 0.876i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.481 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9088222575 + 1.535782108i\)
\(L(\frac12)\) \(\approx\) \(0.9088222575 + 1.535782108i\)
\(L(1)\) \(\approx\) \(1.183158930 + 0.2978944466i\)
\(L(1)\) \(\approx\) \(1.183158930 + 0.2978944466i\)

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

−20.23848224448609564668272331130, −19.50939119161820746589341564042, −19.10770316928171450916442948838, −18.01814192843099997664689937428, −17.550452869258586454202234107667, −16.73406824602683380303229457790, −15.37125015756120826616855566451, −15.20696444185213246958901082574, −14.05774447431941713466416983415, −13.668975140449739570763144209846, −12.61751221434658182298687607903, −12.13655643377795027456338181279, −11.26845479021030008980777830036, −10.035219563678798954672681442309, −9.40826699153006612940489947492, −8.67869454388999409859387945072, −7.65953872794254287389316246439, −7.12628991830353279987968509278, −6.35590065449144288418007507584, −5.203157051075308679940898598880, −4.15405313029660341350486197952, −3.38427836816975307417675817107, −2.09783983982787964423120137591, −1.77612039284292591878662400089, −0.31180784235211478967011604468, 1.007654610731869753012854749217, 2.351795145548166888937783990656, 3.13437714240082654408700886263, 3.8156570118389567583837888830, 4.996119012984327013472525248266, 5.50696013920289717321642867901, 6.798115578751165408710094385071, 7.71486335342948421861330765266, 8.45973337260427626064823823410, 9.13513680999350740588342789847, 10.06324336632089432676104406472, 10.60027150596938170834799205880, 11.55695876881344670224889430225, 12.46728503629232094358027255006, 13.43490556304039460590103718968, 14.11014563829049402105822834598, 14.690573038192030839723360107570, 15.559496483860201831583651083468, 16.354660917296701323226016150975, 16.71112881769904708940804524870, 18.05899933796319132804109517950, 18.56940314736268463327544657443, 19.5915243828690606420336398712, 20.13419947511610656095301216902, 20.74255545261694949282275040010

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