Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $0.999 - 0.0216i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.468 + 0.883i)2-s + (0.725 − 0.687i)3-s + (−0.561 − 0.827i)4-s + (3.92 − 0.863i)5-s + (0.267 + 0.963i)6-s + (−0.604 + 0.459i)7-s + (0.994 − 0.108i)8-s + (0.0541 − 0.998i)9-s + (−1.07 + 3.87i)10-s + (−0.299 − 0.750i)11-s + (−0.976 − 0.214i)12-s + (0.00237 + 0.0438i)13-s + (−0.122 − 0.749i)14-s + (2.25 − 3.32i)15-s + (−0.370 + 0.928i)16-s + (−3.64 − 2.76i)17-s + ⋯
L(s)  = 1  + (−0.331 + 0.624i)2-s + (0.419 − 0.397i)3-s + (−0.280 − 0.413i)4-s + (1.75 − 0.386i)5-s + (0.109 + 0.393i)6-s + (−0.228 + 0.173i)7-s + (0.351 − 0.0382i)8-s + (0.0180 − 0.332i)9-s + (−0.339 + 1.22i)10-s + (−0.0901 − 0.226i)11-s + (−0.281 − 0.0620i)12-s + (0.000659 + 0.0121i)13-s + (−0.0328 − 0.200i)14-s + (0.581 − 0.858i)15-s + (−0.0925 + 0.232i)16-s + (−0.883 − 0.671i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $0.999 - 0.0216i$
motivic weight  =  \(1\)
character  :  $\chi_{354} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :1/2),\ 0.999 - 0.0216i)$
$L(1)$  $\approx$  $1.59529 + 0.0172380i$
$L(\frac12)$  $\approx$  $1.59529 + 0.0172380i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.468 - 0.883i)T \)
3 \( 1 + (-0.725 + 0.687i)T \)
59 \( 1 + (4.69 - 6.08i)T \)
good5 \( 1 + (-3.92 + 0.863i)T + (4.53 - 2.09i)T^{2} \)
7 \( 1 + (0.604 - 0.459i)T + (1.87 - 6.74i)T^{2} \)
11 \( 1 + (0.299 + 0.750i)T + (-7.98 + 7.56i)T^{2} \)
13 \( 1 + (-0.00237 - 0.0438i)T + (-12.9 + 1.40i)T^{2} \)
17 \( 1 + (3.64 + 2.76i)T + (4.54 + 16.3i)T^{2} \)
19 \( 1 + (-5.62 + 1.89i)T + (15.1 - 11.4i)T^{2} \)
23 \( 1 + (6.09 - 3.67i)T + (10.7 - 20.3i)T^{2} \)
29 \( 1 + (-3.97 - 7.49i)T + (-16.2 + 24.0i)T^{2} \)
31 \( 1 + (3.23 + 1.08i)T + (24.6 + 18.7i)T^{2} \)
37 \( 1 + (-6.37 - 0.693i)T + (36.1 + 7.95i)T^{2} \)
41 \( 1 + (-3.43 - 2.06i)T + (19.2 + 36.2i)T^{2} \)
43 \( 1 + (1.90 - 4.77i)T + (-31.2 - 29.5i)T^{2} \)
47 \( 1 + (4.68 + 1.03i)T + (42.6 + 19.7i)T^{2} \)
53 \( 1 + (2.33 + 8.42i)T + (-45.4 + 27.3i)T^{2} \)
61 \( 1 + (1.52 - 2.87i)T + (-34.2 - 50.4i)T^{2} \)
67 \( 1 + (-7.12 + 0.774i)T + (65.4 - 14.4i)T^{2} \)
71 \( 1 + (15.3 + 3.38i)T + (64.4 + 29.8i)T^{2} \)
73 \( 1 + (0.871 + 5.31i)T + (-69.1 + 23.3i)T^{2} \)
79 \( 1 + (0.0990 + 0.0937i)T + (4.27 + 78.8i)T^{2} \)
83 \( 1 + (2.39 - 2.82i)T + (-13.4 - 81.9i)T^{2} \)
89 \( 1 + (6.37 + 12.0i)T + (-49.9 + 73.6i)T^{2} \)
97 \( 1 + (0.451 - 2.75i)T + (-91.9 - 30.9i)T^{2} \)
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

−11.41603989166830031622601855260, −10.10598587625448029232344968707, −9.414991327157004998619333473902, −8.899015615729737966594423482778, −7.68551502766987445127399414622, −6.59600803399952291455513114258, −5.83028085879281474008912526619, −4.88135324919248562629463429028, −2.81631564598653004655007040445, −1.46592831205716371280829495577, 1.85100707612864779402211691789, 2.80218442234827253562076338284, 4.22927660691220114469105949646, 5.62481480372937538232498979099, 6.57974742730182578719569698309, 7.949336074943944417838888422115, 9.050705751026973448166853580348, 9.877474369520741749020914965155, 10.18502196196280249140410645897, 11.16800341869834990441113819681

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