Properties

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

Origins

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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  :  $\chi_{57} (41, \cdot )$
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 \( 1 + (6.91 - 8.24i)T + (-16.8 - 95.5i)T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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