Properties

Degree $2$
Conductor $672$
Sign $-0.0854 + 0.996i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−1.37 − 2.37i)5-s + (2.64 + 0.0585i)7-s + (−0.499 − 0.866i)9-s + (0.771 − 1.33i)11-s + 6.03·13-s − 2.74·15-s + (−3.74 + 6.48i)17-s + (−3.01 − 5.22i)19-s + (1.37 − 2.26i)21-s + (−3.74 − 6.48i)23-s + (−1.27 + 2.20i)25-s − 0.999·27-s + 1.25·29-s + (2.64 − 4.58i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.614 − 1.06i)5-s + (0.999 + 0.0221i)7-s + (−0.166 − 0.288i)9-s + (0.232 − 0.403i)11-s + 1.67·13-s − 0.709·15-s + (−0.908 + 1.57i)17-s + (−0.692 − 1.19i)19-s + (0.299 − 0.493i)21-s + (−0.781 − 1.35i)23-s + (−0.254 + 0.440i)25-s − 0.192·27-s + 0.232·29-s + (0.475 − 0.822i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0854 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0854 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.0854 + 0.996i$
Motivic weight: \(1\)
Character: $\chi_{672} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.0854 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08346 - 1.18032i\)
\(L(\frac12)\) \(\approx\) \(1.08346 - 1.18032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.64 - 0.0585i)T \)
good5 \( 1 + (1.37 + 2.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.771 + 1.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.03T + 13T^{2} \)
17 \( 1 + (3.74 - 6.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.01 + 5.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.74 + 6.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (-2.64 + 4.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.47 + 4.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 3.45T + 43T^{2} \)
47 \( 1 + (-4.74 - 8.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.91 + 3.32i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.77 + 4.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.29 - 12.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.01 - 3.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 + (6.27 - 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.89 + 6.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.52T + 83T^{2} \)
89 \( 1 + (-4.74 - 8.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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

−10.52841055033616791617676288983, −8.697670393192257870508505044952, −8.670178960889794818931316926860, −8.067331257812689711019143189707, −6.68211860277510966361863437767, −5.87065261332527867005514744901, −4.48447240508059446606187707422, −3.94908938491063240132677140897, −2.12498169124565234063860487991, −0.893900328245166901811766946596, 1.83677340753415590168192409366, 3.30740317663983878250352593892, 4.04718165133076204803657424869, 5.14770163838071184466546251330, 6.40131092908350301163045097684, 7.29243465275903447864330703045, 8.191536240029716002371870012593, 8.862160423259817962591067183369, 10.06428188958348668479574524136, 10.75935421743338275700199460932

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