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  :  $\chi_{57} (14, \cdot )$
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

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

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