Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $0.977 - 0.211i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s − 7-s + (−0.499 − 0.866i)9-s + 0.999·12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)28-s − 31-s + (−0.499 + 0.866i)36-s − 37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s − 7-s + (−0.499 − 0.866i)9-s + 0.999·12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.5 + 0.866i)28-s − 31-s + (−0.499 + 0.866i)36-s − 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.977 - 0.211i$
motivic weight  =  \(0\)
character  :  $\chi_{57} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 57,\ (\ :0),\ 0.977 - 0.211i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3735359401\)
\(L(\frac12)\)  \(\approx\)  \(0.3735359401\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;19\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 - T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−15.73698933792598545940443005947, −14.50422369454586197462514135495, −13.52036331503366893986103235457, −12.01208341174308721549737477039, −10.73867704030288327198980522926, −9.755934096435888961389015983965, −8.979906840158385872094533206307, −6.56213261943062567216093815027, −5.42600079494021974315876790764, −3.89339273183383705579882973780, 3.30843370639598070011740501216, 5.50846861495591810506841875830, 7.02046440325267567096761130699, 8.113025414063899120618495471630, 9.530714621167639954748207724266, 11.16295984834053954222603330510, 12.40570419368847873598634726769, 13.04786003793567616204031954427, 13.91118790105346893717398966072, 15.78158863451418310600567398366

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