↑

Properties

Label 2-3920-980.319-c0-0-0
Degree $2$
Conductor $3920$
Sign $0.981 + 0.191i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.108 + 0.277i)3-s + (0.988 + 0.149i)5-s + (−0.563 − 0.826i)7-s + (0.667 − 0.619i)9-s + (0.0663 + 0.290i)15-s + (0.167 − 0.246i)21-s + (0.139 + 1.85i)23-s + (0.955 + 0.294i)25-s + (0.513 + 0.247i)27-s + (−0.134 + 0.0648i)29-s + (−0.433 − 0.900i)35-s + (0.914 − 1.14i)41-s + (−0.848 − 1.06i)43-s + (0.752 − 0.513i)45-s + (1.49 − 0.460i)47-s + ⋯
L(s)  = 1  + (0.108 + 0.277i)3-s + (0.988 + 0.149i)5-s + (−0.563 − 0.826i)7-s + (0.667 − 0.619i)9-s + (0.0663 + 0.290i)15-s + (0.167 − 0.246i)21-s + (0.139 + 1.85i)23-s + (0.955 + 0.294i)25-s + (0.513 + 0.247i)27-s + (−0.134 + 0.0648i)29-s + (−0.433 − 0.900i)35-s + (0.914 − 1.14i)41-s + (−0.848 − 1.06i)43-s + (0.752 − 0.513i)45-s + (1.49 − 0.460i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.981 + 0.191i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.981 + 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.636175994\)
\(L(\frac12)\) \(\approx\) \(1.636175994\)
\(L(1)\) 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 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (0.563 + 0.826i)T \)
good3 \( 1 + (-0.108 - 0.277i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.139 - 1.85i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (-0.914 + 1.14i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.848 + 1.06i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-1.49 + 0.460i)T + (0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (0.603 + 0.411i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.250 + 1.09i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
97 \( 1 - T^{2} \)
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

−9.062543093350377470652665213309, −7.66556548284999867743736492299, −7.13872083259194694446506138878, −6.45389884243583695404385198194, −5.70334451364254183556244617784, −4.89934031332904729706891393720, −3.81922646297672343357056075670, −3.38189811793252841715780981424, −2.13472180850385588251408599982, −1.08340442023049561555738035476, 1.27646060442270484824169514110, 2.38251019163307299780631904933, 2.79832679770748966426259891271, 4.24978058977876880039314317339, 4.96180756225627617500669697233, 5.81949476060539486553642972641, 6.41980446350917189811474096910, 7.05492275729832971350909329945, 8.063573092565035566628758803559, 8.686241899995697142429533831420

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