↑

Properties

Label 2-847-77.48-c0-0-0
Degree $2$
Conductor $847$
Sign $0.0694 - 0.997i$
Analytic cond. $0.422708$
Root an. cond. $0.650160$
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  + (1.30 + 0.951i)2-s + (0.500 + 1.53i)4-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.499 + 1.53i)14-s + (−0.5 − 1.53i)18-s − 1.61·23-s + (0.309 − 0.951i)25-s + (−1.30 + 0.951i)28-s + (0.190 + 0.587i)29-s + 0.999·32-s + (0.499 − 1.53i)36-s + (−0.5 − 1.53i)37-s + 0.618·43-s + ⋯
L(s)  = 1  + (1.30 + 0.951i)2-s + (0.500 + 1.53i)4-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.499 + 1.53i)14-s + (−0.5 − 1.53i)18-s − 1.61·23-s + (0.309 − 0.951i)25-s + (−1.30 + 0.951i)28-s + (0.190 + 0.587i)29-s + 0.999·32-s + (0.499 − 1.53i)36-s + (−0.5 − 1.53i)37-s + 0.618·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.0694 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.857355591\)
\(L(\frac12)\) \(\approx\) \(1.857355591\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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

−10.80328931807353023724585656024, −9.577623060045082676609661936922, −8.607019613574213935048365323715, −7.950500234343849208336061157512, −6.84300404044628132804466699745, −5.98575299678563106505725137823, −5.54451540347844783517538984300, −4.49493466612197636975203312587, −3.50610390342706678757228314155, −2.40396896627239681441679428807, 1.64318609130184251032681965757, 2.81799586484449248407738289829, 3.81371964758773118665333724851, 4.64536468671673748583616658284, 5.47775432361047704952402501558, 6.39076992498300905837987920073, 7.64965076088538736801926289493, 8.414347416921750139149260219695, 9.812177555937591283798709291387, 10.50873127037413216336573654280

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