↑

Properties

Label 2-20-20.3-c7-0-6
Degree $2$
Conductor $20$
Sign $0.434 - 0.900i$
Analytic cond. $6.24770$
Root an. cond. $2.49954$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−8 + 8i)2-s − 128i·4-s + (278 − 29i)5-s + (1.02e3 + 1.02e3i)8-s + 2.18e3i·9-s + (−1.99e3 + 2.45e3i)10-s + (1.10e4 + 1.10e4i)13-s − 1.63e4·16-s + (1.71e4 − 1.71e4i)17-s + (−1.74e4 − 1.74e4i)18-s + (−3.71e3 − 3.55e4i)20-s + (7.64e4 − 1.61e4i)25-s − 1.76e5·26-s + 1.20e5i·29-s + (1.31e5 − 1.31e5i)32-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − i·4-s + (0.994 − 0.103i)5-s + (0.707 + 0.707i)8-s + i·9-s + (−0.629 + 0.776i)10-s + (1.38 + 1.38i)13-s − 16-s + (0.846 − 0.846i)17-s + (−0.707 − 0.707i)18-s + (−0.103 − 0.994i)20-s + (0.978 − 0.206i)25-s − 1.96·26-s + 0.920i·29-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(6.24770\)
Root analytic conductor: \(2.49954\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :7/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.13849 + 0.714704i\)
\(L(\frac12)\) \(\approx\) \(1.13849 + 0.714704i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8 - 8i)T \)
5 \( 1 + (-278 + 29i)T \)
good3 \( 1 - 2.18e3iT^{2} \)
7 \( 1 + 8.23e5iT^{2} \)
11 \( 1 - 1.94e7T^{2} \)
13 \( 1 + (-1.10e4 - 1.10e4i)T + 6.27e7iT^{2} \)
17 \( 1 + (-1.71e4 + 1.71e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 8.93e8T^{2} \)
23 \( 1 - 3.40e9iT^{2} \)
29 \( 1 - 1.20e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.75e10T^{2} \)
37 \( 1 + (-1.57e5 + 1.57e5i)T - 9.49e10iT^{2} \)
41 \( 1 + 8.82e5T + 1.94e11T^{2} \)
43 \( 1 - 2.71e11iT^{2} \)
47 \( 1 + 5.06e11iT^{2} \)
53 \( 1 + (1.40e6 + 1.40e6i)T + 1.17e12iT^{2} \)
59 \( 1 + 2.48e12T^{2} \)
61 \( 1 - 5.32e5T + 3.14e12T^{2} \)
67 \( 1 + 6.06e12iT^{2} \)
71 \( 1 - 9.09e12T^{2} \)
73 \( 1 + (-4.64e6 - 4.64e6i)T + 1.10e13iT^{2} \)
79 \( 1 + 1.92e13T^{2} \)
83 \( 1 - 2.71e13iT^{2} \)
89 \( 1 + 9.56e6iT - 4.42e13T^{2} \)
97 \( 1 + (1.06e7 - 1.06e7i)T - 8.07e13iT^{2} \)
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

−16.75520893545454615585206339526, −16.14817392322647560194652974510, −14.28783265519472665884493908568, −13.51990427076908147332252019516, −11.14628512162971523645375543368, −9.797252429544666142123507030542, −8.532225630042284690305041447421, −6.74964228890560416569568618706, −5.24410925729165942778758030810, −1.64299363056142997136252761803, 1.16754566942834989161443641667, 3.30907914136089690084584111462, 6.15267675561746440316927099331, 8.275227131921386394716489390084, 9.684068590209540765400341746927, 10.76334303112620915233552916864, 12.42070219283652372986892641195, 13.52215987512946742893001613428, 15.32092970610744729844989238605, 16.98878904658663768884702821905

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