↑

Properties

Label 2-40e2-20.7-c1-0-29
Degree $2$
Conductor $1600$
Sign $-0.850 - 0.525i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)3-s + (1 − i)7-s − i·9-s + 6i·11-s + (−1 + i)13-s + (−1 − i)17-s − 4·19-s − 2·21-s + (−5 − 5i)23-s + (−4 + 4i)27-s − 8i·29-s + 2i·31-s + (6 − 6i)33-s + (−5 − 5i)37-s + 2·39-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (0.377 − 0.377i)7-s − 0.333i·9-s + 1.80i·11-s + (−0.277 + 0.277i)13-s + (−0.242 − 0.242i)17-s − 0.917·19-s − 0.436·21-s + (−1.04 − 1.04i)23-s + (−0.769 + 0.769i)27-s − 1.48i·29-s + 0.359i·31-s + (1.04 − 1.04i)33-s + (−0.821 − 0.821i)37-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (5 + 5i)T + 23iT^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (7 - 7i)T - 47iT^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (7 - 7i)T - 67iT^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (9 - 9i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (5 + 5i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-3 - 3i)T + 97iT^{2} \)
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

−9.025637204908989211419191621093, −7.889536571156655130389814405102, −7.28854716051763515581253346644, −6.55940039953861065185824892891, −5.86575297264700713333873134176, −4.51785488697542070280998210141, −4.24887415762888992609219846950, −2.48054409441270760871203712167, −1.56730681119818120057759854084, 0, 1.78081875858630517063994731620, 3.10534604220966593671916779248, 4.05225121294251739274196101591, 5.11131520097096109500924912143, 5.67345537595415015490197509243, 6.39077459417847177487225285081, 7.63282397342565243643138374584, 8.406837679272210433957835242484, 8.948862844657013768046769506814

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