Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.0291 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.448 + 2.54i)2-s + (0.504 − 1.65i)3-s + (−4.37 + 1.59i)4-s + (−0.533 + 1.46i)5-s + (4.43 + 0.538i)6-s + (1.49 − 2.59i)7-s + (−3.42 − 5.93i)8-s + (−2.49 − 1.67i)9-s + (−3.96 − 0.699i)10-s + (1.05 − 0.611i)11-s + (0.432 + 8.05i)12-s + (0.203 − 0.242i)13-s + (7.25 + 2.64i)14-s + (2.16 + 1.62i)15-s + (6.41 − 5.38i)16-s + (−5.00 + 0.882i)17-s + ⋯
L(s)  = 1  + (0.316 + 1.79i)2-s + (0.291 − 0.956i)3-s + (−2.18 + 0.796i)4-s + (−0.238 + 0.655i)5-s + (1.81 + 0.219i)6-s + (0.565 − 0.979i)7-s + (−1.21 − 2.09i)8-s + (−0.830 − 0.556i)9-s + (−1.25 − 0.221i)10-s + (0.319 − 0.184i)11-s + (0.124 + 2.32i)12-s + (0.0564 − 0.0672i)13-s + (1.93 + 0.706i)14-s + (0.558 + 0.419i)15-s + (1.60 − 1.34i)16-s + (−1.21 + 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0291 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0291 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.0291 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 0.0291 - 0.999i)$
$L(1)$  $\approx$  $0.680820 + 0.661248i$
$L(\frac12)$  $\approx$  $0.680820 + 0.661248i$
$L(\frac{3}{2})$   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 \( 1 + (-0.504 + 1.65i)T \)
19 \( 1 + (0.461 - 4.33i)T \)
good2 \( 1 + (-0.448 - 2.54i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.533 - 1.46i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.49 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.05 + 0.611i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.203 + 0.242i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (5.00 - 0.882i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-0.211 - 0.581i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.204 + 1.16i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (4.50 + 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.64iT - 37T^{2} \)
41 \( 1 + (-0.118 + 0.0991i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-8.99 - 3.27i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-8.03 - 1.41i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-2.88 + 1.04i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.468 + 2.65i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.47 - 2.35i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-8.96 - 1.58i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (12.9 + 4.71i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.335 - 0.281i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.940 + 1.12i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-9.46 - 5.46i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.02 + 5.05i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (18.2 - 3.22i)T + (91.1 - 33.1i)T^{2} \)
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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

−15.24598787769030660608086717296, −14.40395766608295190422883528512, −13.79365965909385198508441716796, −12.74528398877480574271040846241, −11.03259986898804825973168957545, −8.935396267977817078411835917273, −7.75258961857716889152217272859, −7.08965394851649268025015528767, −5.98013733297923467174623208320, −4.03065015756233248617212531846, 2.48419763154511860551357595394, 4.25533746471329045740840367005, 5.15939751814214065412083682445, 8.795429182988543433929697451308, 9.103809746490045656224312906612, 10.63069376270207414385349422280, 11.48121859348336131983254137109, 12.41141440792709904896417896154, 13.61043964363685890483144427215, 14.74730821244428315079917712677

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