Properties

Label 2-1400-280.83-c0-0-5
Degree $2$
Conductor $1400$
Sign $0.793 + 0.608i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + 11-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)18-s + (0.965 − 0.258i)22-s + (0.707 − 0.707i)23-s + (−0.965 − 0.258i)28-s − 1.73·29-s + (0.258 − 0.965i)32-s + (0.499 + 0.866i)36-s + (0.707 + 0.707i)37-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + i·9-s + 11-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)18-s + (0.965 − 0.258i)22-s + (0.707 − 0.707i)23-s + (−0.965 − 0.258i)28-s − 1.73·29-s + (0.258 − 0.965i)32-s + (0.499 + 0.866i)36-s + (0.707 + 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.924455902\)
\(L(\frac12)\) \(\approx\) \(1.924455902\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 - iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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

−9.885606556822739165542401933310, −9.036095820558415054602566116362, −7.78698496238020185561543825565, −7.03840942925624014084061344919, −6.37063204274593170195998760935, −5.41728060393882104276126082813, −4.46907491015074442792857825478, −3.73635276484342410361127850386, −2.73043743462125967300242324941, −1.47228903260477238735748361695, 1.78771350263553708087260607097, 3.21234037999579597618764266069, 3.65931690518395841056680973446, 4.83212742528104612060734788157, 5.86075846558359032359825282561, 6.38848536549007603383273534572, 7.09824385048030977965453513134, 8.107033659108864867954027910685, 9.309101440152034169972001935060, 9.463890025577830447473145459291

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