↑

Properties

Label 2-1040-13.10-c1-0-24
Degree $2$
Conductor $1040$
Sign $-0.0183 + 0.999i$
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  + (0.800 + 1.38i)3-s + i·5-s + (−3.75 − 2.16i)7-s + (0.219 − 0.380i)9-s + (−1.5 + 0.866i)11-s + (−3.11 − 1.81i)13-s + (−1.38 + 0.800i)15-s + (3.75 − 6.49i)17-s + (−4.65 − 2.68i)19-s − 6.93i·21-s + (0.580 + 1.00i)23-s − 25-s + 5.50·27-s + (1.01 + 1.75i)29-s − 7.86i·31-s + ⋯
L(s)  = 1  + (0.461 + 0.800i)3-s + 0.447i·5-s + (−1.41 − 0.818i)7-s + (0.0732 − 0.126i)9-s + (−0.452 + 0.261i)11-s + (−0.863 − 0.504i)13-s + (−0.357 + 0.206i)15-s + (0.909 − 1.57i)17-s + (−1.06 − 0.616i)19-s − 1.51i·21-s + (0.121 + 0.209i)23-s − 0.200·25-s + 1.05·27-s + (0.187 + 0.325i)29-s − 1.41i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0183 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0183 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.0183 + 0.999i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.0183 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8031257265\)
\(L(\frac12)\) \(\approx\) \(0.8031257265\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (3.11 + 1.81i)T \)
good3 \( 1 + (-0.800 - 1.38i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.75 + 2.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.75 + 6.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.65 + 2.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.580 - 1.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.01 - 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (8.25 - 4.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.69 + 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.09 - 3.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + (5.49 + 3.17i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.85 - 3.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.55 + 2.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.8 - 6.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.23iT - 73T^{2} \)
79 \( 1 - 8.16T + 79T^{2} \)
83 \( 1 - 0.456iT - 83T^{2} \)
89 \( 1 + (11.4 - 6.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.43 + 1.40i)T + (48.5 + 84.0i)T^{2} \)
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

−9.637591346082512523975092050582, −9.390793796897987711287928813100, −7.987554364224916746890570843433, −7.12032582196107477481885145249, −6.55277407071994883025113381376, −5.23495052533984509502954485656, −4.27652781207890853561533575607, −3.27626644474106446126916519642, −2.71983081724028883539782300687, −0.32266560393930414345194730489, 1.69333485603486307024679840007, 2.66913156882416296424376209294, 3.71633008406674587796381817368, 5.05226446080775046734335532037, 6.08543081326041723276673737503, 6.68705064360246997630608344122, 7.77667537581240208717972783345, 8.427192904085258782026577054056, 9.176437636695050074090366428644, 10.07628595456207182596041890311

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