Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $-0.235 + 0.971i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.886 − 0.322i)2-s + (−0.858 − 1.50i)3-s + (−0.850 − 0.713i)4-s + (−0.485 − 0.578i)5-s + (0.275 + 1.61i)6-s + (1.38 − 2.39i)7-s + (1.46 + 2.54i)8-s + (−1.52 + 2.58i)9-s + (0.243 + 0.669i)10-s + (2.28 − 1.31i)11-s + (−0.343 + 1.89i)12-s + (1.90 + 0.335i)13-s + (−1.99 + 1.67i)14-s + (−0.453 + 1.22i)15-s + (−0.0947 − 0.537i)16-s + (−0.220 + 0.606i)17-s + ⋯
L(s)  = 1  + (−0.626 − 0.228i)2-s + (−0.495 − 0.868i)3-s + (−0.425 − 0.356i)4-s + (−0.217 − 0.258i)5-s + (0.112 + 0.657i)6-s + (0.522 − 0.905i)7-s + (0.518 + 0.898i)8-s + (−0.508 + 0.860i)9-s + (0.0770 + 0.211i)10-s + (0.687 − 0.396i)11-s + (−0.0991 + 0.546i)12-s + (0.527 + 0.0930i)13-s + (−0.534 + 0.448i)14-s + (−0.117 + 0.317i)15-s + (−0.0236 − 0.134i)16-s + (−0.0535 + 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $-0.235 + 0.971i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ -0.235 + 0.971i)$
$L(1)$  $\approx$  $0.320681 - 0.407729i$
$L(\frac12)$  $\approx$  $0.320681 - 0.407729i$
$L(\frac{3}{2})$   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 \( 1 + (0.858 + 1.50i)T \)
19 \( 1 + (1.94 - 3.90i)T \)
good2 \( 1 + (0.886 + 0.322i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.485 + 0.578i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.38 + 2.39i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.28 + 1.31i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.90 - 0.335i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.220 - 0.606i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.61 + 6.69i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (8.33 - 3.03i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.10 - 1.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.09iT - 37T^{2} \)
41 \( 1 + (-0.930 - 5.27i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.12 - 0.947i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-3.48 - 9.56i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.16 - 3.49i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (7.06 + 2.57i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.37 + 1.15i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.35 + 9.20i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.08 + 1.74i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.952 - 5.39i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-9.22 + 1.62i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (7.02 + 4.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.233 - 1.32i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.64 + 12.7i)T + (-74.3 - 62.3i)T^{2} \)
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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

−14.58542391820631647213219202406, −13.79987907792019109100219222124, −12.64032223884178547877118357588, −11.23092037033763777985424034217, −10.54381145175646232180093398758, −8.817898865930612983946972670165, −7.85413037305440271472375412557, −6.29257120143955330646551314430, −4.59487718742301936808130048149, −1.21369979428468023501115898218, 3.82645612123430162176176594497, 5.36927908803365504022633701681, 7.17751451054438603809819777835, 8.835858482971626779606640943395, 9.397570903658132460639261253317, 10.96375211123774369806694689136, 11.86831999586555205025389284498, 13.29263518874126178343733817565, 14.97375360044389710831562305346, 15.51433604562960849297239436575

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