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Properties

Label 1-1400-1400.1061-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.637 - 0.770i$
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  + (0.104 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.584405692 - 0.7460905478i\)
\(L(\frac12)\) \(\approx\) \(1.584405692 - 0.7460905478i\)
\(L(1)\) \(\approx\) \(1.147482616 - 0.3609083224i\)
\(L(1)\) \(\approx\) \(1.147482616 - 0.3609083224i\)

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

−20.95527371445342485570649877252, −20.2492254322785363488639535332, −19.44410528960214216086104761544, −18.862552460096893061162572135796, −17.64684805335160170306132614118, −16.95681418040130291395369583771, −16.45749955425374366276166503201, −15.55885660090483114680831619579, −14.84007599481208221253177147397, −14.17937058517525560270075092698, −13.53295832419463316673469629489, −12.16139962817688469219204666045, −11.70239877973068797676708744951, −10.831568800012830890026338114248, −9.94895288540632004624573346399, −9.26731493000138597843154978170, −8.73844747429571204485924921841, −7.59426648494475355601745788792, −6.65736847423752788212056106616, −5.80847567089263423771604763371, −4.61509637020792669469224531010, −4.34465043836123049422703067503, −3.15223602032232955084109090153, −2.37585400152876238469158395978, −0.9176904722617413257839057115, 0.95816193564954571443272364631, 1.59672076854374913551261078214, 2.94204850852860686090883174942, 3.484959798856746768515740158736, 4.86355161224012640559453728868, 5.85978570609025327696066558514, 6.424197508687834874402833670043, 7.4572545074092183156406102547, 8.04173780644016711386676293098, 8.84285605205724398600894260410, 9.82657354046303531316613613032, 10.67838088032383646889969388837, 11.80261585761867867995759492991, 12.16355125464895191307256954674, 13.03282535174161656308630501993, 13.775020505130431328536102646766, 14.590191319494115035064528224935, 15.0824453322232629449406307242, 16.40316853448175700686787595583, 17.001941955434508312298445157412, 17.74037407585056118221835316210, 18.431537629950620907246579970464, 19.30594588995157936111832640905, 19.71099918227779677405427813448, 20.52610920123570394101227924234

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