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  :  $\chi_{57} (41, \cdot )$
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

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

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