↑

Properties

Label 2-921-921.422-c0-0-0
Degree $2$
Conductor $921$
Sign $0.232 + 0.972i$
Analytic cond. $0.459638$
Root an. cond. $0.677966$
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  + (0.445 − 0.895i)3-s + (−0.273 − 0.961i)4-s + (1.09 + 0.995i)7-s + (−0.602 − 0.798i)9-s + (−0.982 − 0.183i)12-s + (1.67 − 1.03i)13-s + (−0.850 + 0.526i)16-s + (−1.58 + 0.981i)19-s + (1.37 − 0.533i)21-s + (−0.850 − 0.526i)25-s + (−0.982 + 0.183i)27-s + (0.658 − 1.32i)28-s + (0.658 + 1.32i)31-s + (−0.602 + 0.798i)36-s + (−0.404 − 0.368i)37-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)3-s + (−0.273 − 0.961i)4-s + (1.09 + 0.995i)7-s + (−0.602 − 0.798i)9-s + (−0.982 − 0.183i)12-s + (1.67 − 1.03i)13-s + (−0.850 + 0.526i)16-s + (−1.58 + 0.981i)19-s + (1.37 − 0.533i)21-s + (−0.850 − 0.526i)25-s + (−0.982 + 0.183i)27-s + (0.658 − 1.32i)28-s + (0.658 + 1.32i)31-s + (−0.602 + 0.798i)36-s + (−0.404 − 0.368i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.232 + 0.972i$
Analytic conductor: \(0.459638\)
Root analytic conductor: \(0.677966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :0),\ 0.232 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.190208357\)
\(L(\frac12)\) \(\approx\) \(1.190208357\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.445 + 0.895i)T \)
307 \( 1 + (-0.445 + 0.895i)T \)
good2 \( 1 + (0.273 + 0.961i)T^{2} \)
5 \( 1 + (0.850 + 0.526i)T^{2} \)
7 \( 1 + (-1.09 - 0.995i)T + (0.0922 + 0.995i)T^{2} \)
11 \( 1 + (0.602 - 0.798i)T^{2} \)
13 \( 1 + (-1.67 + 1.03i)T + (0.445 - 0.895i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (1.58 - 0.981i)T + (0.445 - 0.895i)T^{2} \)
23 \( 1 + (0.273 - 0.961i)T^{2} \)
29 \( 1 + (0.273 + 0.961i)T^{2} \)
31 \( 1 + (-0.658 - 1.32i)T + (-0.602 + 0.798i)T^{2} \)
37 \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \)
41 \( 1 + (-0.0922 + 0.995i)T^{2} \)
43 \( 1 + (0.510 + 0.197i)T + (0.739 + 0.673i)T^{2} \)
47 \( 1 + (-0.445 + 0.895i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.982 - 0.183i)T^{2} \)
61 \( 1 + (1.83 + 0.710i)T + (0.739 + 0.673i)T^{2} \)
67 \( 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)T^{2} \)
71 \( 1 + (-0.739 + 0.673i)T^{2} \)
73 \( 1 + (1.25 - 0.778i)T + (0.445 - 0.895i)T^{2} \)
79 \( 1 + (-0.831 + 0.322i)T + (0.739 - 0.673i)T^{2} \)
83 \( 1 + (-0.0922 + 0.995i)T^{2} \)
89 \( 1 + (-0.932 - 0.361i)T^{2} \)
97 \( 1 + (-1.67 - 1.03i)T + (0.445 + 0.895i)T^{2} \)
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

−10.22351313042589477010918400388, −8.858255963752836178225125482056, −8.525962912307244898253650163354, −7.87971126729762622908851164178, −6.34167926225830115021949317323, −5.99418318151804013060274935097, −5.03638427904939204094975655624, −3.69478998974286792896497560918, −2.20286165979589085298595334775, −1.36956007134155558216193233615, 2.00049920662982963816672551321, 3.46958577795780580749108403991, 4.27342075032947547423950310233, 4.62224225546637218213897561832, 6.19844260960570679118240861875, 7.32740547605439025329518706201, 8.211805743497416334379892773980, 8.651432321155960628071197562899, 9.452996744509925053234297266732, 10.60927439877564012852971556471

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