↑

Properties

Label 2-3920-980.879-c0-0-1
Degree $2$
Conductor $3920$
Sign $-0.747 + 0.664i$
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  + (0.0841 − 1.12i)3-s + (−0.826 − 0.563i)5-s + (0.680 − 0.733i)7-s + (−0.266 − 0.0401i)9-s + (−0.702 + 0.880i)15-s + (−0.766 − 0.825i)21-s + (1.90 − 0.587i)23-s + (0.365 + 0.930i)25-s + (0.183 − 0.802i)27-s + (−0.425 − 1.86i)29-s + (−0.974 + 0.222i)35-s + (−1.78 + 0.858i)41-s + (0.268 + 0.129i)43-s + (0.197 + 0.183i)45-s + (0.317 − 0.807i)47-s + ⋯
L(s)  = 1  + (0.0841 − 1.12i)3-s + (−0.826 − 0.563i)5-s + (0.680 − 0.733i)7-s + (−0.266 − 0.0401i)9-s + (−0.702 + 0.880i)15-s + (−0.766 − 0.825i)21-s + (1.90 − 0.587i)23-s + (0.365 + 0.930i)25-s + (0.183 − 0.802i)27-s + (−0.425 − 1.86i)29-s + (−0.974 + 0.222i)35-s + (−1.78 + 0.858i)41-s + (0.268 + 0.129i)43-s + (0.197 + 0.183i)45-s + (0.317 − 0.807i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.747 + 0.664i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220080174\)
\(L(\frac12)\) \(\approx\) \(1.220080174\)
\(L(1)\) 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 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (-0.680 + 0.733i)T \)
good3 \( 1 + (-0.0841 + 1.12i)T + (-0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.955 + 0.294i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.90 + 0.587i)T + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.425 + 1.86i)T + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (1.78 - 0.858i)T + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.268 - 0.129i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.317 + 0.807i)T + (-0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (-0.109 + 0.101i)T + (0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.848 - 1.06i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (1.88 + 0.284i)T + (0.955 + 0.294i)T^{2} \)
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

−8.293207852861102993244344151594, −7.53126639435231959071776247446, −7.14585173770273298807947351096, −6.41498475307270832242652556337, −5.26750287067979062554118835345, −4.55856245732343064483367486355, −3.84477434594320840059343480948, −2.68873401220981575876325767849, −1.54491638646472071669363188992, −0.74601939175638316501931667118, 1.59776606738271285330756499025, 3.03084950728226653752429010919, 3.41057704506870742883952226897, 4.48858983318313775992428622991, 4.98357428656727388234390896491, 5.73064926958508770240998886321, 7.00170892012236748333127816890, 7.32348402825157724335293517092, 8.468304075178220324938172310304, 8.855546279030120178003171577972

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