Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $-0.0746 - 0.997i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.49 + 1.25i)2-s + (1.40 + 1.01i)3-s + (0.317 − 1.79i)4-s + (−0.487 + 0.0858i)5-s + (−3.37 + 0.254i)6-s + (−0.969 + 1.67i)7-s + (−0.170 − 0.295i)8-s + (0.957 + 2.84i)9-s + (0.621 − 0.741i)10-s + (3.99 − 2.30i)11-s + (2.26 − 2.20i)12-s + (−1.79 − 4.92i)13-s + (−0.658 − 3.73i)14-s + (−0.771 − 0.371i)15-s + (4.05 + 1.47i)16-s + (1.61 + 1.92i)17-s + ⋯
L(s)  = 1  + (−1.05 + 0.889i)2-s + (0.812 + 0.583i)3-s + (0.158 − 0.898i)4-s + (−0.217 + 0.0384i)5-s + (−1.37 + 0.103i)6-s + (−0.366 + 0.634i)7-s + (−0.0602 − 0.104i)8-s + (0.319 + 0.947i)9-s + (0.196 − 0.234i)10-s + (1.20 − 0.695i)11-s + (0.653 − 0.637i)12-s + (−0.497 − 1.36i)13-s + (−0.176 − 0.998i)14-s + (−0.199 − 0.0958i)15-s + (1.01 + 0.369i)16-s + (0.392 + 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0746 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0746 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $-0.0746 - 0.997i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ -0.0746 - 0.997i)$
$L(1)$  $\approx$  $0.436115 + 0.469961i$
$L(\frac12)$  $\approx$  $0.436115 + 0.469961i$
$L(\frac{3}{2})$   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 \( 1 + (-1.40 - 1.01i)T \)
19 \( 1 + (-2.80 + 3.33i)T \)
good2 \( 1 + (1.49 - 1.25i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.487 - 0.0858i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.969 - 1.67i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.99 + 2.30i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.79 + 4.92i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.61 - 1.92i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (3.35 + 0.591i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.872 + 0.731i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.31 + 0.758i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.28iT - 37T^{2} \)
41 \( 1 + (9.40 + 3.42i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.829 + 4.70i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-3.82 + 4.56i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.71 - 9.73i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.172 + 0.144i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.90 - 10.7i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.266 + 0.317i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.540 + 3.06i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (0.118 + 0.0430i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-4.94 + 13.5i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.05 - 3.49i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.32 + 1.93i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (4.49 + 5.36i)T + (-16.8 + 95.5i)T^{2} \)
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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

−15.48284657135154227927176574168, −15.04081790606790627596934158426, −13.64492669422895213007740239530, −12.09894774834151752849460817035, −10.33309250608360902920251727959, −9.334543850863952500925282715168, −8.493613335929007690407328437502, −7.44155771886603118057388267855, −5.79397010900304721672926137852, −3.43969106937715938386105244166, 1.75617596536792581522094399510, 3.76310103476034690224379768737, 6.81284578832639633389425805930, 7.965257617654942528614897831033, 9.390273304204249155431213296689, 9.824369186031762104600022982340, 11.66683682181331007572492981084, 12.25317305543349101227605886159, 13.91611923438182563186208863328, 14.62170980166879236961585184713

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