Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $-0.941 - 0.336i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.79·2-s + (0.128 − 0.222i)3-s + 1.20·4-s + (−1.10 + 1.90i)5-s + (−0.230 + 0.398i)6-s + (1.71 + 2.97i)7-s + 1.42·8-s + (1.46 + 2.54i)9-s + (1.97 − 3.41i)10-s − 5.96·11-s + (0.154 − 0.268i)12-s + (1.79 + 3.11i)13-s + (−3.06 − 5.31i)14-s + (0.283 + 0.491i)15-s − 4.95·16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 1.26·2-s + (0.0742 − 0.128i)3-s + 0.602·4-s + (−0.492 + 0.853i)5-s + (−0.0940 + 0.162i)6-s + (0.648 + 1.12i)7-s + 0.503·8-s + (0.488 + 0.846i)9-s + (0.623 − 1.08i)10-s − 1.79·11-s + (0.0447 − 0.0774i)12-s + (0.499 + 0.864i)13-s + (−0.820 − 1.42i)14-s + (0.0732 + 0.126i)15-s − 1.23·16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-0.941 - 0.336i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -0.941 - 0.336i)$
$L(1)$  $\approx$  $0.0806784 + 0.464977i$
$L(\frac12)$  $\approx$  $0.0806784 + 0.464977i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (5.22 + 3.96i)T \)
good2 \( 1 + 1.79T + 2T^{2} \)
3 \( 1 + (-0.128 + 0.222i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.10 - 1.90i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.71 - 2.97i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.96T + 11T^{2} \)
13 \( 1 + (-1.79 - 3.11i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (0.138 - 0.239i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.74 + 4.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 + 4.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.48 + 2.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.22 - 5.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.04T + 41T^{2} \)
47 \( 1 - 9.75T + 47T^{2} \)
53 \( 1 + (1.89 - 3.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 + (-3.11 - 5.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.60 - 9.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.74 + 8.22i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.76 + 8.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.80 - 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.77 - 3.07i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0943 - 0.163i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.63060836016673950410144144481, −10.04780848818464797982001854429, −8.806081352727937345020668655808, −8.261357123631762566944780156255, −7.61037163268249195634843786289, −6.82491556355167701620555164699, −5.41895971632951919305727383360, −4.48366775490638692275552747906, −2.71776179019599787505835948528, −1.86282486646295257403174017194, 0.39638532225986932825774913314, 1.39135799921448533823738855378, 3.37727170150896615769732464652, 4.56649511646218260420346545187, 5.31157488368922636234793710430, 7.07505195944657044673583311397, 7.67074641074003447809223317366, 8.310232496756510302476085586415, 9.026938377865845908230618878819, 10.09241069509953764633572962870

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