Properties

Label 2-1400-5.4-c1-0-22
Degree $2$
Conductor $1400$
Sign $-0.447 + 0.894i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·7-s + 3·9-s − 4·11-s − 2i·13-s − 6i·17-s − 8·19-s − 6·29-s + 8·31-s − 2i·37-s + 2·41-s + 4i·43-s − 8i·47-s − 49-s − 6i·53-s − 6·61-s + ⋯
L(s)  = 1  − 0.377i·7-s + 9-s − 1.20·11-s − 0.554i·13-s − 1.45i·17-s − 1.83·19-s − 1.11·29-s + 1.43·31-s − 0.328i·37-s + 0.312·41-s + 0.609i·43-s − 1.16i·47-s − 0.142·49-s − 0.824i·53-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.026896981\)
\(L(\frac12)\) \(\approx\) \(1.026896981\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 - 3T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.410916468241825229766145265603, −8.396385774894016392694733449916, −7.62987080601243796752423280141, −7.00781810551954968235323917709, −6.02224124161427072590741799890, −4.94747425022579641938684617095, −4.32377305846473301833053232353, −3.08334817946026054783807946778, −2.03056861323365657171738892390, −0.39778913001841796688175152024, 1.64670023248576994942418186967, 2.59723362413832936647261406193, 4.01828761299910274334162031536, 4.61039410653294145651867066362, 5.81206519022109189319313440131, 6.47758094124440516270583930495, 7.47011285572364215902889192047, 8.243724306143163975356336755463, 8.919330382079513576548056115159, 9.992903094702103705351172268204

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