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Properties

Label 1-1400-1400.1389-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.986 + 0.165i$
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  + (0.104 − 0.994i)3-s + (−0.978 − 0.207i)9-s + (0.978 − 0.207i)11-s + (−0.309 − 0.951i)13-s + (0.913 + 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.669 − 0.743i)23-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.913 − 0.406i)31-s + (−0.104 − 0.994i)33-s + (−0.978 − 0.207i)37-s + (−0.978 + 0.207i)39-s + (−0.309 − 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)3-s + (−0.978 − 0.207i)9-s + (0.978 − 0.207i)11-s + (−0.309 − 0.951i)13-s + (0.913 + 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.669 − 0.743i)23-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.913 − 0.406i)31-s + (−0.104 − 0.994i)33-s + (−0.978 − 0.207i)37-s + (−0.978 + 0.207i)39-s + (−0.309 − 0.951i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.986 + 0.165i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1232875478 - 1.477821571i\)
\(L(\frac12)\) \(\approx\) \(-0.1232875478 - 1.477821571i\)
\(L(1)\) \(\approx\) \(0.9078673044 - 0.5398343277i\)
\(L(1)\) \(\approx\) \(0.9078673044 - 0.5398343277i\)

Euler product

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

−21.02100198886966868783276328531, −20.40064095085843949397339386906, −19.4892032624159211316729422676, −19.00149579354857776005600603453, −17.80672590744891671182397305669, −17.06987588685942787163575397294, −16.39861102314792193037313307848, −15.860313328716985774217613329599, −14.76524620366379482136680945301, −14.32454370628063448439567429910, −13.70869124275488549158077851466, −12.230013865698403972318093043518, −11.85342002863472395540038831728, −10.95095426365810723206967102889, −9.92221491687365519695355650625, −9.57133503693070185340079142065, −8.69063756516189849252816088021, −7.81341241532657108322470524473, −6.77797982072312574142304016880, −5.84603617853545852526352987664, −5.023546213415184530589530002078, −4.037227523415301900510839609541, −3.56776748235226499218660884468, −2.3487377990623585873751270643, −1.27328551333200613663105847953, 0.30013925347993526947003823779, 1.1095678918007826486771171941, 2.17026344519395758682996935731, 3.07428496883108166560659996075, 3.99388702801540708149529630588, 5.30780036263899641200541311189, 5.99159823001529718943385792556, 6.90441968186813849713242982431, 7.53739955424619804965991435697, 8.493305456971181766114102539389, 9.04793999074224041826886502729, 10.22760670272576102657382129772, 11.00299938097429535841592425968, 12.059986097579049831089249842801, 12.414638774458594203695562192144, 13.279407149905500766114912775689, 14.17839817215308738918214407138, 14.611711038381764042812396036257, 15.60877674874734296526845184286, 16.63014279696339148673230047214, 17.37697682098464066988954080183, 17.85730230001157252333644756510, 18.835829752398737565200392145638, 19.38645262543772192205188380092, 20.09388995222693010855294642768

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