↑

Properties

Label 2-672-168.5-c1-0-10
Degree $2$
Conductor $672$
Sign $0.277 - 0.960i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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  + (0.618 + 1.61i)3-s + (2.46 + 1.42i)5-s + (1.02 − 2.44i)7-s + (−2.23 + 2.00i)9-s + (2.42 + 4.20i)11-s − 2.75·13-s + (−0.780 + 4.87i)15-s + (1.75 + 3.03i)17-s + (3.14 − 5.44i)19-s + (4.58 + 0.142i)21-s + (−3.15 − 1.82i)23-s + (1.56 + 2.71i)25-s + (−4.61 − 2.38i)27-s + 3.90·29-s + (0.858 − 0.495i)31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s + (1.10 + 0.637i)5-s + (0.385 − 0.922i)7-s + (−0.745 + 0.666i)9-s + (0.731 + 1.26i)11-s − 0.763·13-s + (−0.201 + 1.25i)15-s + (0.425 + 0.736i)17-s + (0.721 − 1.24i)19-s + (0.999 + 0.0309i)21-s + (−0.658 − 0.380i)23-s + (0.313 + 0.542i)25-s + (−0.888 − 0.458i)27-s + 0.725·29-s + (0.154 − 0.0889i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65592 + 1.24504i\)
\(L(\frac12)\) \(\approx\) \(1.65592 + 1.24504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.618 - 1.61i)T \)
7 \( 1 + (-1.02 + 2.44i)T \)
good5 \( 1 + (-2.46 - 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.42 - 4.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + (-1.75 - 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.14 + 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.90T + 29T^{2} \)
31 \( 1 + (-0.858 + 0.495i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.06 + 0.614i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.10T + 41T^{2} \)
43 \( 1 - 5.11iT - 43T^{2} \)
47 \( 1 + (5.61 - 9.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.00 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.890 + 0.514i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.24 - 2.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 2.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.75iT - 71T^{2} \)
73 \( 1 + (-0.291 + 0.168i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.80 + 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.138iT - 83T^{2} \)
89 \( 1 + (-0.580 + 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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

−10.40708783995543547667432391551, −9.831933399736611970229932190734, −9.389526871305294235913287784395, −8.083355341964108223816311204802, −7.14070998978745261935631291804, −6.26235115171377151476697234950, −4.96401318624425245835036821780, −4.31409441610290866736515803539, −3.00359928707636466139863226735, −1.85536189558085881709789248179, 1.20803548025406834320073434226, 2.23060705451592418350548573165, 3.41282109709940638608595157270, 5.28989877475866888230823331950, 5.73681644681083997608718085937, 6.66541835388966216506439711214, 7.908766863050566870276441634085, 8.589248589279400882915938197169, 9.322418376618440166655895669394, 10.03889633211666162388997857399

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