↑

Properties

Label 2-3920-140.83-c0-0-0
Degree $2$
Conductor $3920$
Sign $-0.865 - 0.501i$
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  + (−0.382 + 0.923i)5-s − i·9-s + (−1.30 + 1.30i)13-s + (−0.541 − 0.541i)17-s + (−0.707 − 0.707i)25-s + 1.41i·29-s + (−1.41 + 1.41i)37-s + 1.84i·41-s + (0.923 + 0.382i)45-s + (−1 − i)53-s − 0.765i·61-s + (−0.707 − 1.70i)65-s + (−0.541 + 0.541i)73-s − 81-s + (0.707 − 0.292i)85-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)5-s − i·9-s + (−1.30 + 1.30i)13-s + (−0.541 − 0.541i)17-s + (−0.707 − 0.707i)25-s + 1.41i·29-s + (−1.41 + 1.41i)37-s + 1.84i·41-s + (0.923 + 0.382i)45-s + (−1 − i)53-s − 0.765i·61-s + (−0.707 − 1.70i)65-s + (−0.541 + 0.541i)73-s − 81-s + (0.707 − 0.292i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.865 - 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4858231709\)
\(L(\frac12)\) \(\approx\) \(0.4858231709\)
\(L(1)\) 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 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 \)
good3 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
17 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.765iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 0.765T + T^{2} \)
97 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
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

−9.012630393855166331167768383478, −8.215792803674233635057347095560, −7.21056640490637271893770257602, −6.81280070774284042049618729705, −6.34203854242738806821476338874, −5.05678914336009139205767829226, −4.42385626864748815714567047855, −3.43088530351153404625137716452, −2.79055536947196672104633573995, −1.66954332889816148973458914029, 0.25515624994551923654888876164, 1.83402563255309843185158657600, 2.66725119416108736071871952287, 3.87354357405061775161264392689, 4.61064563405625941240034101077, 5.34214540694665673058057062088, 5.82799173530573509205237776248, 7.19385012514864520032068609670, 7.64033885625493520185345875911, 8.292046439178275321369684333409

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