Properties

Label 2-640-5.4-c1-0-18
Degree $2$
Conductor $640$
Sign $0.241 + 0.970i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
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  − 1.70i·3-s + (2.17 − 0.539i)5-s − 2.63i·7-s + 0.0783·9-s + 5.41·11-s + 6.34i·13-s + (−0.921 − 3.70i)15-s − 3.41i·17-s − 3.26·19-s − 4.49·21-s + 1.36i·23-s + (4.41 − 2.34i)25-s − 5.26i·27-s − 2·29-s − 4.68·31-s + ⋯
L(s)  = 1  − 0.986i·3-s + (0.970 − 0.241i)5-s − 0.994i·7-s + 0.0261·9-s + 1.63·11-s + 1.75i·13-s + (−0.237 − 0.957i)15-s − 0.829i·17-s − 0.748·19-s − 0.981·21-s + 0.285i·23-s + (0.883 − 0.468i)25-s − 1.01i·27-s − 0.371·29-s − 0.840·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.241 + 0.970i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.241 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50686 - 1.17828i\)
\(L(\frac12)\) \(\approx\) \(1.50686 - 1.17828i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.539i)T \)
good3 \( 1 + 1.70iT - 3T^{2} \)
7 \( 1 + 2.63iT - 7T^{2} \)
11 \( 1 - 5.41T + 11T^{2} \)
13 \( 1 - 6.34iT - 13T^{2} \)
17 \( 1 + 3.41iT - 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 - 1.36iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 - 5.75iT - 37T^{2} \)
41 \( 1 + 7.75T + 41T^{2} \)
43 \( 1 - 4.44iT - 43T^{2} \)
47 \( 1 + 4.78iT - 47T^{2} \)
53 \( 1 - 1.65iT - 53T^{2} \)
59 \( 1 - 3.26T + 59T^{2} \)
61 \( 1 + 2.49T + 61T^{2} \)
67 \( 1 + 7.86iT - 67T^{2} \)
71 \( 1 + 6.15T + 71T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 14.9iT - 83T^{2} \)
89 \( 1 - 8.52T + 89T^{2} \)
97 \( 1 - 4.58iT - 97T^{2} \)
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

−10.28418202924388796173496659025, −9.400878632271644383817764608259, −8.844551501760473184137579360482, −7.45515996808005722640010121383, −6.67198336643281702045211255376, −6.38843382021056645443949287934, −4.76528246467753403098928198957, −3.85168577274616627528014900771, −1.97456931574189927582470939468, −1.25678848664019354551084836957, 1.76714006595715220593271578996, 3.15912371916785394519871354883, 4.17698500981928025281958118525, 5.46364143948528066999107001517, 5.96138699379922525405491832147, 7.06644250512636940827475457903, 8.591187303622340888009112286047, 9.053013229049960321478545216173, 9.991395470807294461559911605241, 10.48809798627960219600247196959

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