Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.997 - 0.0646i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.745 + 0.625i)2-s + (0.0227 − 1.73i)3-s + (−0.182 + 1.03i)4-s + (3.79 − 0.668i)5-s + (1.06 + 1.30i)6-s + (−0.469 + 0.813i)7-s + (−1.48 − 2.57i)8-s + (−2.99 − 0.0789i)9-s + (−2.40 + 2.87i)10-s + (−3.44 + 1.99i)11-s + (1.79 + 0.340i)12-s + (−0.353 − 0.970i)13-s + (−0.158 − 0.901i)14-s + (−1.07 − 6.58i)15-s + (0.740 + 0.269i)16-s + (0.804 + 0.958i)17-s + ⋯
L(s)  = 1  + (−0.527 + 0.442i)2-s + (0.0131 − 0.999i)3-s + (−0.0913 + 0.518i)4-s + (1.69 − 0.298i)5-s + (0.435 + 0.533i)6-s + (−0.177 + 0.307i)7-s + (−0.525 − 0.909i)8-s + (−0.999 − 0.0263i)9-s + (−0.761 + 0.907i)10-s + (−1.03 + 0.600i)11-s + (0.516 + 0.0981i)12-s + (−0.0979 − 0.269i)13-s + (−0.0424 − 0.240i)14-s + (−0.276 − 1.69i)15-s + (0.185 + 0.0673i)16-s + (0.195 + 0.232i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.997 - 0.0646i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 0.997 - 0.0646i)$
$L(1)$  $\approx$  $0.752142 + 0.0243217i$
$L(\frac12)$  $\approx$  $0.752142 + 0.0243217i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (-0.0227 + 1.73i)T \)
19 \( 1 + (4.22 - 1.09i)T \)
good2 \( 1 + (0.745 - 0.625i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-3.79 + 0.668i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.469 - 0.813i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.44 - 1.99i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.353 + 0.970i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.804 - 0.958i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.477 - 0.0842i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.32 + 2.79i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-4.26 - 2.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.10iT - 37T^{2} \)
41 \( 1 + (2.04 + 0.745i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.00 + 5.68i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.96 + 5.91i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.34 - 7.60i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-5.26 + 4.41i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.740 + 4.19i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.918 + 1.09i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.438 + 2.48i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-8.28 - 3.01i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (1.50 - 4.12i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3.66 - 2.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.56 - 2.75i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (6.91 + 8.24i)T + (-16.8 + 95.5i)T^{2} \)
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

−15.33815943906876514476904918940, −13.84153608934125669909046272436, −12.95602643933962346507748373240, −12.39446016659406824477569125268, −10.31253276706278026752767350073, −9.133155658137541700741982231298, −8.059536609143553624012683588058, −6.71241508658591204233581585497, −5.57481108667172655730027824333, −2.38270525398588913453961998960, 2.56207921107022638495579854103, 5.15037944379188205793096949226, 6.16061506480817802602742977908, 8.688332631782797584892628433426, 9.730233301008090417395339116466, 10.34729716360187648883185241093, 11.15275347054492846082042707525, 13.28972213459351100389050163580, 14.16347240811914020412538276146, 15.09231346072462553354643257150

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