↑

Properties

Label 2-40e2-20.3-c1-0-25
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 + (−3 − 3i)7-s + i·9-s + 2i·11-s + (3 + 3i)13-s + (−1 + i)17-s + 4·19-s + 6·21-s + (−1 + i)23-s + (−4 − 4i)27-s − 10i·31-s + (−2 − 2i)33-s + (−1 + i)37-s − 6·39-s − 10·41-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (−1.13 − 1.13i)7-s + 0.333i·9-s + 0.603i·11-s + (0.832 + 0.832i)13-s + (−0.242 + 0.242i)17-s + 0.917·19-s + 1.30·21-s + (−0.208 + 0.208i)23-s + (−0.769 − 0.769i)27-s − 1.79i·31-s + (−0.348 − 0.348i)33-s + (−0.164 + 0.164i)37-s − 0.960·39-s − 1.56·41-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: $\chi_{1600} (1343, \cdot )$
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 \( 1 + (3 + 3i)T + 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + (3 + 3i)T + 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (5 - 5i)T - 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
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

−9.492720387548640946458888795814, −8.238087443664100648297461376930, −7.35466130558282302581545567147, −6.60444173453214833013871216286, −5.89290166053203571505447211701, −4.76596156559381746265679850396, −4.08010266609314751459190029639, −3.26524033774252583616087916396, −1.68082192265665586254738887024, 0, 1.38134933259869946030630000976, 3.00867318863603356950809490039, 3.43677035799624185483612676385, 5.14296477111565930907404971565, 5.81728526750977657581662811718, 6.40693536831449581400013557028, 7.08197431740022071042206041486, 8.283539223919954088699060381263, 8.912927045451480904730248988512

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