Properties

Label 2-1400-5.4-c1-0-19
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·3-s + i·7-s − 9-s − 2i·17-s + 2·19-s + 2·21-s − 8i·23-s − 4i·27-s − 2·29-s + 4·31-s − 6i·37-s − 2·41-s − 8i·43-s − 4i·47-s − 49-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.377i·7-s − 0.333·9-s − 0.485i·17-s + 0.458·19-s + 0.436·21-s − 1.66i·23-s − 0.769i·27-s − 0.371·29-s + 0.718·31-s − 0.986i·37-s − 0.312·41-s − 1.21i·43-s − 0.583i·47-s − 0.142·49-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

\(L(1)\) \(\approx\) \(1.508546354\)
\(L(\frac12)\) \(\approx\) \(1.508546354\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.144724936973781732491592415782, −8.397889481519338765926418803948, −7.60097980278152930322486802212, −6.89597773728039066374595571624, −6.19706853742635374274902779089, −5.25351275791154087860041031210, −4.18472634125489876184966099370, −2.82993948735686424815583918179, −1.96510182166407410189717775949, −0.64455666675166078320055723254, 1.45497257569858995093002737719, 3.10451445644428243907641366325, 3.84425622099164970912232649079, 4.71208934799557050061407574633, 5.46989344098252350302431261500, 6.50027241114960682451324655593, 7.49480057458445876057315126169, 8.267844583602414184539988046573, 9.325804699386767276234996236196, 9.782137192299039015420428127973

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