↑

Properties

Label 2-40e2-8.5-c1-0-7
Degree $2$
Conductor $1600$
Sign $-0.707 - 0.707i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.44i·3-s + 0.898·9-s − 0.550i·11-s − 7.89·17-s + 8.34i·19-s + 5.65i·27-s + 0.797·33-s + 12.7·41-s + 10i·43-s − 7·49-s − 11.4i·51-s − 12.1·57-s + 6i·59-s + 14.3i·67-s − 13.6·73-s + ⋯
L(s)  = 1  + 0.836i·3-s + 0.299·9-s − 0.165i·11-s − 1.91·17-s + 1.91i·19-s + 1.08i·27-s + 0.138·33-s + 1.99·41-s + 1.52i·43-s − 49-s − 1.60i·51-s − 1.60·57-s + 0.781i·59-s + 1.75i·67-s − 1.60·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.236371791\)
\(L(\frac12)\) \(\approx\) \(1.236371791\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.44iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 0.550iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7.89T + 17T^{2} \)
19 \( 1 - 8.34iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 12.7T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 11.4iT - 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−9.712777573139476191852720978586, −9.049499471334203509275259918242, −8.228509010537515459924623850771, −7.34967322594072463007182652433, −6.37979273506460949545321030173, −5.61092136861478168337222489920, −4.42653700757017734927518446000, −4.08664443615262597745045310805, −2.86483721348894094557734912271, −1.56787672600461043390312226791, 0.46999951498999245449660171786, 1.91519532285394232707928963630, 2.71185581064950569311684051030, 4.19537219932644720003695052693, 4.84258022410827206373677875929, 6.09127159967759435054672510756, 6.89635390597825684759990802923, 7.24312141000043432129456574263, 8.317289485323344187913193582376, 9.052170775609728684513233151156

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