Properties

Label 2-8280-69.68-c1-0-40
Degree $2$
Conductor $8280$
Sign $0.999 - 0.0350i$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
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  − 5-s − 3.13i·7-s − 2.04·11-s + 5.04·13-s + 2.17·17-s − 2.11i·19-s + (−4.01 − 2.62i)23-s + 25-s + 8.97i·29-s + 3.32·31-s + 3.13i·35-s + 2.50i·37-s + 8.54i·41-s + 10.3i·43-s + 11.6i·47-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.18i·7-s − 0.618·11-s + 1.39·13-s + 0.526·17-s − 0.485i·19-s + (−0.836 − 0.548i)23-s + 0.200·25-s + 1.66i·29-s + 0.597·31-s + 0.529i·35-s + 0.412i·37-s + 1.33i·41-s + 1.57i·43-s + 1.69i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.999 - 0.0350i$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8280} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 0.999 - 0.0350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.764093196\)
\(L(\frac12)\) \(\approx\) \(1.764093196\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
23 \( 1 + (4.01 + 2.62i)T \)
good7 \( 1 + 3.13iT - 7T^{2} \)
11 \( 1 + 2.04T + 11T^{2} \)
13 \( 1 - 5.04T + 13T^{2} \)
17 \( 1 - 2.17T + 17T^{2} \)
19 \( 1 + 2.11iT - 19T^{2} \)
29 \( 1 - 8.97iT - 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 - 2.50iT - 37T^{2} \)
41 \( 1 - 8.54iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 - 2.92T + 53T^{2} \)
59 \( 1 - 0.993iT - 59T^{2} \)
61 \( 1 - 8.28iT - 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 - 9.03T + 73T^{2} \)
79 \( 1 + 5.58iT - 79T^{2} \)
83 \( 1 + 7.01T + 83T^{2} \)
89 \( 1 - 7.84T + 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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

−7.964002375610561356909874999097, −7.15907769604313412658628353079, −6.46256165751861558216170378024, −5.87873841982149926084771843946, −4.75690768555951728152607420348, −4.37841964301721628478850223688, −3.42500972773010560258883586582, −2.95350556285035729435004687511, −1.50964829566534507806063994740, −0.77695782052713540013368993445, 0.58318523999300411432005790094, 1.91099324810051277857200576573, 2.56059715114246017222771818354, 3.68830953995695383809296858600, 3.98209415533555420073411364926, 5.32916762855262783408630389207, 5.61406311776620712452621174266, 6.28505806908705257005880070557, 7.17544129545986919063346795369, 7.982666636052642795224391852608

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