↑

Properties

Label 2-3920-980.919-c0-0-0
Degree $2$
Conductor $3920$
Sign $0.0106 - 0.999i$
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  + (0.432 − 0.400i)3-s + (−0.955 − 0.294i)5-s + (−0.930 − 0.365i)7-s + (−0.0487 + 0.650i)9-s + (−0.531 + 0.255i)15-s + (−0.548 + 0.215i)21-s + (−1.34 + 0.202i)23-s + (0.826 + 0.563i)25-s + (0.607 + 0.761i)27-s + (−1.23 + 1.54i)29-s + (0.781 + 0.623i)35-s + (0.0332 + 0.145i)41-s + (−0.443 + 1.94i)43-s + (0.238 − 0.607i)45-s + (1.61 − 1.09i)47-s + ⋯
L(s)  = 1  + (0.432 − 0.400i)3-s + (−0.955 − 0.294i)5-s + (−0.930 − 0.365i)7-s + (−0.0487 + 0.650i)9-s + (−0.531 + 0.255i)15-s + (−0.548 + 0.215i)21-s + (−1.34 + 0.202i)23-s + (0.826 + 0.563i)25-s + (0.607 + 0.761i)27-s + (−1.23 + 1.54i)29-s + (0.781 + 0.623i)35-s + (0.0332 + 0.145i)41-s + (−0.443 + 1.94i)43-s + (0.238 − 0.607i)45-s + (1.61 − 1.09i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.0106 - 0.999i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (2879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.0106 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5781250466\)
\(L(\frac12)\) \(\approx\) \(0.5781250466\)
\(L(1)\) 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 \( 1 + (0.955 + 0.294i)T \)
7 \( 1 + (0.930 + 0.365i)T \)
good3 \( 1 + (-0.432 + 0.400i)T + (0.0747 - 0.997i)T^{2} \)
11 \( 1 + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.955 - 0.294i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.34 - 0.202i)T + (0.955 - 0.294i)T^{2} \)
29 \( 1 + (1.23 - 1.54i)T + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.733 - 0.680i)T^{2} \)
41 \( 1 + (-0.0332 - 0.145i)T + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.443 - 1.94i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (-1.61 + 1.09i)T + (0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.733 + 0.680i)T^{2} \)
59 \( 1 + (-0.826 + 0.563i)T^{2} \)
61 \( 1 + (-0.535 - 1.36i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.67 - 0.807i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.147 - 1.97i)T + (-0.988 - 0.149i)T^{2} \)
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

−8.726056659989161596684175754240, −8.000067057215880929527204503513, −7.42738420145384108192989917451, −6.89652571302258447013854175249, −5.90465220790198298427883344728, −5.03699379308732806920975375504, −4.08450621603714206459421067867, −3.45541000573524930688993752100, −2.55983971302906819852375119131, −1.35610296393124404804941663452, 0.31060988468450885352514815116, 2.25033047597719963572542880504, 3.11056680766412560481155128988, 3.88606080429917378376768151127, 4.25128446990568595531707342064, 5.66220368115590164228690438693, 6.22179450489793495858789351825, 7.08656857478117971377638806683, 7.73337596940491864425480223201, 8.608147204039902153967016230330

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