Properties

Label 1-1400-1400.411-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.103 + 0.994i$
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  + (−0.913 + 0.406i)3-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (0.309 + 0.951i)13-s + (0.104 + 0.994i)17-s + (−0.913 − 0.406i)19-s + (0.978 − 0.207i)23-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.913 − 0.406i)33-s + (−0.669 + 0.743i)37-s + (−0.669 − 0.743i)39-s + (−0.309 − 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)3-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (0.309 + 0.951i)13-s + (0.104 + 0.994i)17-s + (−0.913 − 0.406i)19-s + (0.978 − 0.207i)23-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.104 − 0.994i)31-s + (−0.913 − 0.406i)33-s + (−0.669 + 0.743i)37-s + (−0.669 − 0.743i)39-s + (−0.309 − 0.951i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.103 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7067627599 + 0.7840823941i\)
\(L(\frac12)\) \(\approx\) \(0.7067627599 + 0.7840823941i\)
\(L(1)\) \(\approx\) \(0.8102658907 + 0.2551172370i\)
\(L(1)\) \(\approx\) \(0.8102658907 + 0.2551172370i\)

Euler product

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

−20.73429755711734275979296370655, −19.575560378258747550459820975511, −19.10585109260723329532660544760, −18.22156650815130114264706749928, −17.58825815597303348742762518143, −16.88732370464107046441974973177, −16.16935986074138357184975108183, −15.48031805231532222040709029511, −14.40256574443365254632548473562, −13.61766522630058119636660534094, −12.840196015971738532738947617412, −12.14784483188231225218630054216, −11.31898915918814669614646278616, −10.74220435856783481044934512369, −9.91205258364589935544712271104, −8.81499654941366399695749731081, −8.03273133455318032283360922328, −7.04409200218371640577976941912, −6.37488089372890902330149674021, −5.552902400622969018640668252976, −4.833244425930974321071221862130, −3.72037897415714039807128631650, −2.70730513027741633911279767447, −1.42703015833519534855468207322, −0.54934611170248141879745193153, 1.12229043395502086398616914724, 2.06257271110019875052750410921, 3.53122698234943080517156081161, 4.35894272198670788555555265013, 4.9049371872005959863000900640, 6.18849019307094211888853092360, 6.54879867065487570801141368193, 7.482325935479106711723692023000, 8.81822074825554390867572478055, 9.31054452048077399572555557220, 10.38341264076164897460478873827, 10.89455598425705496703039113475, 11.79282518628633453098559216416, 12.42374398167509504381104069830, 13.16759560936543585998683818558, 14.32896895192284177474679660334, 15.027400633813559065028103755353, 15.70516377631985874463983723991, 16.64252979937760273353814834313, 17.21389600977972890655358676455, 17.68023332864511232254537202939, 18.85065412834101157455647993304, 19.27235478082757206147863378734, 20.44656293657908138600588076008, 21.105392586684732864943001032911

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