↑

Properties

Label 2-836-11.9-c1-0-16
Degree $2$
Conductor $836$
Sign $-0.605 + 0.795i$
Analytic cond. $6.67549$
Root an. cond. $2.58369$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.22 − 1.61i)3-s + (−0.682 − 2.10i)5-s + (−1.20 − 0.878i)7-s + (1.41 − 4.36i)9-s + (−0.809 − 3.21i)11-s + (−0.525 + 1.61i)13-s + (−4.92 − 3.57i)15-s + (0.606 + 1.86i)17-s + (−0.809 + 0.587i)19-s − 4.11·21-s − 1.66·23-s + (0.100 − 0.0727i)25-s + (−1.34 − 4.15i)27-s + (−0.395 − 0.287i)29-s + (−0.633 + 1.95i)31-s + ⋯
L(s)  = 1  + (1.28 − 0.934i)3-s + (−0.305 − 0.939i)5-s + (−0.456 − 0.331i)7-s + (0.472 − 1.45i)9-s + (−0.243 − 0.969i)11-s + (−0.145 + 0.448i)13-s + (−1.27 − 0.922i)15-s + (0.147 + 0.452i)17-s + (−0.185 + 0.134i)19-s − 0.897·21-s − 0.346·23-s + (0.0200 − 0.0145i)25-s + (−0.259 − 0.798i)27-s + (−0.0734 − 0.0533i)29-s + (−0.113 + 0.350i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(836\)    =    \(2^{2} \cdot 11 \cdot 19\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(6.67549\)
Root analytic conductor: \(2.58369\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{836} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 836,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869750 - 1.75439i\)
\(L(\frac12)\) \(\approx\) \(0.869750 - 1.75439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (0.809 + 3.21i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (-2.22 + 1.61i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.682 + 2.10i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.20 + 0.878i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.525 - 1.61i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.606 - 1.86i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + 1.66T + 23T^{2} \)
29 \( 1 + (0.395 + 0.287i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.633 - 1.95i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.335 + 0.244i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.82 + 3.50i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.80T + 43T^{2} \)
47 \( 1 + (-4.64 + 3.37i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.08 + 6.41i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.40 + 3.19i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.83 - 8.73i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.0250T + 67T^{2} \)
71 \( 1 + (-0.549 - 1.69i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.88 - 6.45i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.86 - 5.75i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.01 + 3.13i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.67T + 89T^{2} \)
97 \( 1 + (-4.25 + 13.1i)T + (-78.4 - 57.0i)T^{2} \)
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

−9.609271852267828335615400278631, −8.730987161428012083713164230155, −8.376276218187631500814004898842, −7.52117986033866270303970207703, −6.68064755622926668733902651095, −5.58650069901090483018515242161, −4.18497517876923526546191602646, −3.32729125193862933749577440279, −2.14174518453237260037225447768, −0.825110463221491727375361531530, 2.39059736041828756392051259462, 3.01247695156295486587768116429, 3.95506308281541936948876312640, 4.90365309498633141263267892215, 6.23745103884592328882853204848, 7.40323125820891591722982810261, 7.88784703198310548759207188711, 9.086960923627150045230567957973, 9.556020309459719291335346775840, 10.35920444294735652228436708439

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