↑

Properties

Label 2-848-212.135-c0-0-0
Degree $2$
Conductor $848$
Sign $0.661 + 0.750i$
Analytic cond. $0.423207$
Root an. cond. $0.650543$
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  + (0.317 − 1.28i)5-s + (0.970 + 0.239i)9-s + (−0.850 − 0.753i)13-s + (−0.627 + 0.329i)17-s + (−0.671 − 0.352i)25-s + (1.10 − 1.59i)29-s + (−0.213 + 1.75i)37-s + (0.764 − 0.527i)41-s + (0.616 − 1.17i)45-s + (−0.748 + 0.663i)49-s + (−0.120 + 0.992i)53-s + (−0.222 + 0.423i)61-s + (−1.24 + 0.855i)65-s + (0.922 + 1.75i)73-s + (0.885 + 0.464i)81-s + ⋯
L(s)  = 1  + (0.317 − 1.28i)5-s + (0.970 + 0.239i)9-s + (−0.850 − 0.753i)13-s + (−0.627 + 0.329i)17-s + (−0.671 − 0.352i)25-s + (1.10 − 1.59i)29-s + (−0.213 + 1.75i)37-s + (0.764 − 0.527i)41-s + (0.616 − 1.17i)45-s + (−0.748 + 0.663i)49-s + (−0.120 + 0.992i)53-s + (−0.222 + 0.423i)61-s + (−1.24 + 0.855i)65-s + (0.922 + 1.75i)73-s + (0.885 + 0.464i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(848\)    =    \(2^{4} \cdot 53\)
Sign: $0.661 + 0.750i$
Analytic conductor: \(0.423207\)
Root analytic conductor: \(0.650543\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{848} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 848,\ (\ :0),\ 0.661 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054456430\)
\(L(\frac12)\) \(\approx\) \(1.054456430\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
53 \( 1 + (0.120 - 0.992i)T \)
good3 \( 1 + (-0.970 - 0.239i)T^{2} \)
5 \( 1 + (-0.317 + 1.28i)T + (-0.885 - 0.464i)T^{2} \)
7 \( 1 + (0.748 - 0.663i)T^{2} \)
11 \( 1 + (0.354 - 0.935i)T^{2} \)
13 \( 1 + (0.850 + 0.753i)T + (0.120 + 0.992i)T^{2} \)
17 \( 1 + (0.627 - 0.329i)T + (0.568 - 0.822i)T^{2} \)
19 \( 1 + (0.120 + 0.992i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1.10 + 1.59i)T + (-0.354 - 0.935i)T^{2} \)
31 \( 1 + (-0.354 - 0.935i)T^{2} \)
37 \( 1 + (0.213 - 1.75i)T + (-0.970 - 0.239i)T^{2} \)
41 \( 1 + (-0.764 + 0.527i)T + (0.354 - 0.935i)T^{2} \)
43 \( 1 + (0.970 - 0.239i)T^{2} \)
47 \( 1 + (-0.885 + 0.464i)T^{2} \)
59 \( 1 + (-0.885 + 0.464i)T^{2} \)
61 \( 1 + (0.222 - 0.423i)T + (-0.568 - 0.822i)T^{2} \)
67 \( 1 + (0.120 - 0.992i)T^{2} \)
71 \( 1 + (-0.970 + 0.239i)T^{2} \)
73 \( 1 + (-0.922 - 1.75i)T + (-0.568 + 0.822i)T^{2} \)
79 \( 1 + (-0.748 - 0.663i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.56 - 0.822i)T + (0.568 - 0.822i)T^{2} \)
97 \( 1 + (1.88 + 0.464i)T + (0.885 + 0.464i)T^{2} \)
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

−9.999902130751496308725482009212, −9.622971188499778223288634889284, −8.516498772604402011724466225933, −7.916406617767148092627924151041, −6.86216045959145977285135788944, −5.78475192346891681028353947044, −4.80496751935132840432927298550, −4.26675595412350298880743888322, −2.57911604990966114381460780923, −1.23556899800798632167287434247, 1.92858228079137076760727379774, 2.98736292078514918660251098569, 4.15150207422697385715899726372, 5.16606776334376746100786546367, 6.63179497091193581911463682777, 6.81162842077853880818513447046, 7.72828401860928655828264427666, 9.061928860195570700698488604430, 9.723124152837444042462381964667, 10.53180278629757267403540798769

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