↑

Properties

Label 2-1040-65.47-c1-0-23
Degree $2$
Conductor $1040$
Sign $0.990 - 0.134i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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 + i)3-s + (2 − i)5-s + 2·7-s + i·9-s + (1 − i)11-s + (2 − 3i)13-s + (−1 + 3i)15-s + (−1 + i)17-s + (3 − 3i)19-s + (−2 + 2i)21-s + (−1 − i)23-s + (3 − 4i)25-s + (−4 − 4i)27-s + 8i·29-s + (−1 − i)31-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (0.894 − 0.447i)5-s + 0.755·7-s + 0.333i·9-s + (0.301 − 0.301i)11-s + (0.554 − 0.832i)13-s + (−0.258 + 0.774i)15-s + (−0.242 + 0.242i)17-s + (0.688 − 0.688i)19-s + (−0.436 + 0.436i)21-s + (−0.208 − 0.208i)23-s + (0.600 − 0.800i)25-s + (−0.769 − 0.769i)27-s + 1.48i·29-s + (−0.179 − 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.990 - 0.134i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771841183\)
\(L(\frac12)\) \(\approx\) \(1.771841183\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
13 \( 1 + (-2 + 3i)T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 + (1 + i)T + 31iT^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (7 + 7i)T + 41iT^{2} \)
43 \( 1 + (1 + i)T + 43iT^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (-9 - 9i)T + 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + (5 + 5i)T + 71iT^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (-11 - 11i)T + 89iT^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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

−10.14270587080169082443167746094, −9.087137622755323864098457333040, −8.493915870847531530810395120950, −7.48422969583501624961889910482, −6.29870182968291452155306811500, −5.40119533790229489367842510617, −5.02701262208173165225212366479, −3.89271663717379464718842348369, −2.42497372166466164169491189817, −1.09890903022930003044714089875, 1.24087826438721230777347148708, 2.15518095014858741026724343543, 3.64844216811107268698501303834, 4.82348358580404945025425857462, 5.89527287074411596659745698973, 6.39595384945979422203430927905, 7.23141041349045793084767261005, 8.143558487344459696991614498479, 9.299927267119627538801175866966, 9.783675720916880866307779596830

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