Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
Sign $0.857 + 0.513i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 − 0.697i)2-s + (1.22 − 1.22i)3-s + (3.02 − 2.61i)4-s + (−5.24 + 5.24i)5-s + (1.44 − 3.14i)6-s − 5.32·7-s + (3.85 − 7.01i)8-s − 2.99i·9-s + (−6.17 + 13.4i)10-s + (12.2 + 12.2i)11-s + (0.507 − 6.90i)12-s + (−5.73 − 5.73i)13-s + (−9.98 + 3.71i)14-s + 12.8i·15-s + (2.33 − 15.8i)16-s − 23.3·17-s + ⋯
L(s)  = 1  + (0.937 − 0.348i)2-s + (0.408 − 0.408i)3-s + (0.757 − 0.653i)4-s + (−1.04 + 1.04i)5-s + (0.240 − 0.524i)6-s − 0.761·7-s + (0.481 − 0.876i)8-s − 0.333i·9-s + (−0.617 + 1.34i)10-s + (1.11 + 1.11i)11-s + (0.0423 − 0.575i)12-s + (−0.441 − 0.441i)13-s + (−0.713 + 0.265i)14-s + 0.856i·15-s + (0.146 − 0.989i)16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.513i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $0.857 + 0.513i$
motivic weight  =  \(2\)
character  :  $\chi_{48} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :1),\ 0.857 + 0.513i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.65617 - 0.457865i\)
\(L(\frac12)\)  \(\approx\)  \(1.65617 - 0.457865i\)
\(L(2)\)   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 \( 1 + (-1.87 + 0.697i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
good5 \( 1 + (5.24 - 5.24i)T - 25iT^{2} \)
7 \( 1 + 5.32T + 49T^{2} \)
11 \( 1 + (-12.2 - 12.2i)T + 121iT^{2} \)
13 \( 1 + (5.73 + 5.73i)T + 169iT^{2} \)
17 \( 1 + 23.3T + 289T^{2} \)
19 \( 1 + (-11.7 + 11.7i)T - 361iT^{2} \)
23 \( 1 - 5.80T + 529T^{2} \)
29 \( 1 + (-18.3 - 18.3i)T + 841iT^{2} \)
31 \( 1 + 16.9iT - 961T^{2} \)
37 \( 1 + (-15.3 + 15.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 29.2iT - 1.68e3T^{2} \)
43 \( 1 + (-33.4 - 33.4i)T + 1.84e3iT^{2} \)
47 \( 1 + 18.2iT - 2.20e3T^{2} \)
53 \( 1 + (66.9 - 66.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (27.1 + 27.1i)T + 3.48e3iT^{2} \)
61 \( 1 + (-65.2 - 65.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (37.6 - 37.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 42.6T + 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + 21.2iT - 6.24e3T^{2} \)
83 \( 1 + (-24.1 + 24.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 52.8iT - 7.92e3T^{2} \)
97 \( 1 + 21.0T + 9.40e3T^{2} \)
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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

−15.10700366629264439080578999824, −14.29012281502386743278538904176, −12.96220445385298483648862829827, −12.00491840895790898239092124506, −10.97324011609666625040783275742, −9.511334292228899367970647727642, −7.28707758726511848799234937687, −6.63535586026509442723145163868, −4.19890698947182026530310114144, −2.80832534364437408566934433913, 3.52608016002707217453223146512, 4.63841283494658027264344300370, 6.48310952935779175703933109029, 8.121062887412855279674195823742, 9.182502270890562886825474041324, 11.29062680877747774680897316354, 12.19119679289400497228183444248, 13.33093332012915598089630550911, 14.38622189110549986402688963545, 15.70566566389284270361654439212

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