Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.821 + 0.570i$
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 + (1.09 − 1.34i)3-s + (−0.182 + 1.03i)4-s + (−3.79 + 0.668i)5-s + (−0.0221 − 1.68i)6-s + (−0.469 + 0.813i)7-s + (1.48 + 2.57i)8-s + (−0.598 − 2.93i)9-s + (−2.40 + 2.87i)10-s + (3.44 − 1.99i)11-s + (1.18 + 1.38i)12-s + (−0.353 − 0.970i)13-s + (0.158 + 0.901i)14-s + (−3.25 + 5.81i)15-s + (0.740 + 0.269i)16-s + (−0.804 − 0.958i)17-s + ⋯
L(s)  = 1  + (0.527 − 0.442i)2-s + (0.632 − 0.774i)3-s + (−0.0913 + 0.518i)4-s + (−1.69 + 0.298i)5-s + (−0.00905 − 0.688i)6-s + (−0.177 + 0.307i)7-s + (0.525 + 0.909i)8-s + (−0.199 − 0.979i)9-s + (−0.761 + 0.907i)10-s + (1.03 − 0.600i)11-s + (0.343 + 0.398i)12-s + (−0.0979 − 0.269i)13-s + (0.0424 + 0.240i)14-s + (−0.841 + 1.50i)15-s + (0.185 + 0.0673i)16-s + (−0.195 − 0.232i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.821 + 0.570i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 0.821 + 0.570i)$
$L(1)$  $\approx$  $1.00017 - 0.313320i$
$L(\frac12)$  $\approx$  $1.00017 - 0.313320i$
$L(\frac{3}{2})$   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 \( 1 + (-1.09 + 1.34i)T \)
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

−14.86036177640490283508033973678, −13.96401895132747954097437652140, −12.60749711792443812435576862752, −12.02991697490941012482122704134, −11.12762428149140010281909131877, −8.739976614003834073886907081007, −8.000826290697523718378470133666, −6.75798124013537162424019600619, −4.11360234924180208293182806047, −3.05437170889461401330356583545, 3.97175359149696364371862271458, 4.57533662717537470428190240860, 6.76902985254044016241760085869, 8.136974956325077170454378385563, 9.422241979189123808409556571398, 10.71985049205457766411654248707, 11.99522279543726530385163985466, 13.40642318207116159801467409173, 14.67676927380734026700343793409, 15.18163399572293079157516030686

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