Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $-0.926 - 0.377i$
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 + (−1.71 − 0.208i)3-s + (−4.37 − 1.59i)4-s + (0.533 + 1.46i)5-s + (1.30 − 4.27i)6-s + (1.49 + 2.59i)7-s + (3.42 − 5.93i)8-s + (2.91 + 0.718i)9-s + (−3.96 + 0.699i)10-s + (−1.05 − 0.611i)11-s + (7.19 + 3.65i)12-s + (0.203 + 0.242i)13-s + (−7.25 + 2.64i)14-s + (−0.611 − 2.63i)15-s + (6.41 + 5.38i)16-s + (5.00 + 0.882i)17-s + ⋯
L(s)  = 1  + (−0.316 + 1.79i)2-s + (−0.992 − 0.120i)3-s + (−2.18 − 0.796i)4-s + (0.238 + 0.655i)5-s + (0.531 − 1.74i)6-s + (0.565 + 0.979i)7-s + (1.21 − 2.09i)8-s + (0.970 + 0.239i)9-s + (−1.25 + 0.221i)10-s + (−0.319 − 0.184i)11-s + (2.07 + 1.05i)12-s + (0.0564 + 0.0672i)13-s + (−1.93 + 0.706i)14-s + (−0.157 − 0.679i)15-s + (1.60 + 1.34i)16-s + (1.21 + 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $-0.926 - 0.377i$
motivic weight  =  \(1\)
character  :  $\chi_{57} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ -0.926 - 0.377i)$
$L(1)$  $\approx$  $0.106192 + 0.541850i$
$L(\frac12)$  $\approx$  $0.106192 + 0.541850i$
$L(\frac{3}{2})$   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 \( 1 + (1.71 + 0.208i)T \)
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

−15.77336849482110010126681912702, −14.96715815844120475050195885163, −13.96814487255184166961561510127, −12.54231199773375099298562313808, −11.01445252062223578015083996931, −9.641559301698530415959726652818, −8.224400669751236275443126175880, −7.00853641558403413815221628306, −5.92010486073338477614544809961, −5.02823642553981213917431043993, 1.24989845106979644859933641810, 3.98278496552045537555988837595, 5.24705229827421399527019873408, 7.79431688863958159466188449306, 9.479426118183436210127666191522, 10.40259665858944823560734641715, 11.18609449908987496089063892264, 12.33036101055787444863588626219, 12.98414375374066248287019895642, 14.28645191572357166097725588648

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