Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
Sign $-0.141 - 0.989i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.125 + 1.99i)2-s + (1.22 + 1.22i)3-s + (−3.96 − 0.500i)4-s + (3.32 + 3.32i)5-s + (−2.59 + 2.29i)6-s − 4.04·7-s + (1.49 − 7.85i)8-s + 2.99i·9-s + (−7.05 + 6.22i)10-s + (6.82 − 6.82i)11-s + (−4.24 − 5.47i)12-s + (4.29 − 4.29i)13-s + (0.506 − 8.06i)14-s + 8.14i·15-s + (15.4 + 3.97i)16-s + 30.1·17-s + ⋯
L(s)  = 1  + (−0.0626 + 0.998i)2-s + (0.408 + 0.408i)3-s + (−0.992 − 0.125i)4-s + (0.665 + 0.665i)5-s + (−0.433 + 0.381i)6-s − 0.577·7-s + (0.187 − 0.982i)8-s + 0.333i·9-s + (−0.705 + 0.622i)10-s + (0.620 − 0.620i)11-s + (−0.353 − 0.456i)12-s + (0.330 − 0.330i)13-s + (0.0361 − 0.576i)14-s + 0.543i·15-s + (0.968 + 0.248i)16-s + 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $-0.141 - 0.989i$
motivic weight  =  \(2\)
character  :  $\chi_{48} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :1),\ -0.141 - 0.989i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.771356 + 0.889311i\)
\(L(\frac12)\)  \(\approx\)  \(0.771356 + 0.889311i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.125 - 1.99i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
good5 \( 1 + (-3.32 - 3.32i)T + 25iT^{2} \)
7 \( 1 + 4.04T + 49T^{2} \)
11 \( 1 + (-6.82 + 6.82i)T - 121iT^{2} \)
13 \( 1 + (-4.29 + 4.29i)T - 169iT^{2} \)
17 \( 1 - 30.1T + 289T^{2} \)
19 \( 1 + (19.7 + 19.7i)T + 361iT^{2} \)
23 \( 1 + 28.2T + 529T^{2} \)
29 \( 1 + (21.3 - 21.3i)T - 841iT^{2} \)
31 \( 1 - 38.0iT - 961T^{2} \)
37 \( 1 + (42.8 + 42.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (-32.6 + 32.6i)T - 1.84e3iT^{2} \)
47 \( 1 + 15.8iT - 2.20e3T^{2} \)
53 \( 1 + (0.476 + 0.476i)T + 2.80e3iT^{2} \)
59 \( 1 + (-9.97 + 9.97i)T - 3.48e3iT^{2} \)
61 \( 1 + (-37.9 + 37.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (-20.0 - 20.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 40.0T + 5.04e3T^{2} \)
73 \( 1 - 30.8iT - 5.32e3T^{2} \)
79 \( 1 - 130. iT - 6.24e3T^{2} \)
83 \( 1 + (2.26 + 2.26i)T + 6.88e3iT^{2} \)
89 \( 1 + 72.2iT - 7.92e3T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−15.77987812980426514565064547600, −14.48046588580079961057182034693, −14.01215264998878059901163350957, −12.68170440624641657090595952197, −10.56702436935985127800263921617, −9.590278315726681588219066947751, −8.401838176818770097639112220975, −6.80831588622261272543734793293, −5.64177409392442631817642587643, −3.54826847181597847311203393806, 1.74188252212475400268684983584, 3.85987524989755973398501719249, 5.88035740005531551202570862876, 8.018242416930068133620961705904, 9.362556412246055969598063405800, 10.07921593900380815152938618213, 11.95713945366780694678000460527, 12.68489516092536826282926810767, 13.70117321108920930880684165015, 14.67072726491174516786435011415

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