Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 7 $
Sign $0.935 + 0.353i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.54i·5-s − 2.64i·7-s − 4.48·11-s − 1.54i·13-s − 23.6·17-s + 24.8·19-s + 35.2i·23-s + 22.5·25-s + 22.4i·29-s + 46.7i·31-s − 4.10·35-s − 58.5i·37-s + 26.9·41-s + 17.1·43-s − 36.1i·47-s + ⋯
L(s)  = 1  − 0.309i·5-s − 0.377i·7-s − 0.407·11-s − 0.119i·13-s − 1.39·17-s + 1.30·19-s + 1.53i·23-s + 0.903·25-s + 0.774i·29-s + 1.50i·31-s − 0.117·35-s − 1.58i·37-s + 0.657·41-s + 0.399·43-s − 0.768i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.935 + 0.353i$
motivic weight  =  \(2\)
character  :  $\chi_{2016} (1135, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2016,\ (\ :1),\ 0.935 + 0.353i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.848524541\)
\(L(\frac12)\)  \(\approx\)  \(1.848524541\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + 2.64iT \)
good5 \( 1 + 1.54iT - 25T^{2} \)
11 \( 1 + 4.48T + 121T^{2} \)
13 \( 1 + 1.54iT - 169T^{2} \)
17 \( 1 + 23.6T + 289T^{2} \)
19 \( 1 - 24.8T + 361T^{2} \)
23 \( 1 - 35.2iT - 529T^{2} \)
29 \( 1 - 22.4iT - 841T^{2} \)
31 \( 1 - 46.7iT - 961T^{2} \)
37 \( 1 + 58.5iT - 1.36e3T^{2} \)
41 \( 1 - 26.9T + 1.68e3T^{2} \)
43 \( 1 - 17.1T + 1.84e3T^{2} \)
47 \( 1 + 36.1iT - 2.20e3T^{2} \)
53 \( 1 + 97.8iT - 2.80e3T^{2} \)
59 \( 1 - 61.5T + 3.48e3T^{2} \)
61 \( 1 + 37.6iT - 3.72e3T^{2} \)
67 \( 1 - 33.3T + 4.48e3T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 - 69.3T + 5.32e3T^{2} \)
79 \( 1 + 38.7iT - 6.24e3T^{2} \)
83 \( 1 - 3.61T + 6.88e3T^{2} \)
89 \( 1 + 44.0T + 7.92e3T^{2} \)
97 \( 1 - 96.1T + 9.40e3T^{2} \)
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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

−8.965972566976297667802747652928, −8.189521173731705043665092345393, −7.22859161976739210769165117238, −6.81233295357438995856908384951, −5.47523711090579121812834759080, −5.06637021885440238671822895099, −3.93546265980307474265407173445, −3.09288602277313798291250248819, −1.86761538093128545670378044296, −0.69154894167603822063880637661, 0.75524526519699635039338133128, 2.31013698089092948857975947442, 2.89554814856293800843005706157, 4.21695805374762711099339458163, 4.88833389682195114279930006533, 5.96921217836367218054909379179, 6.58138070069528642596737400786, 7.47679564536432607867933917393, 8.238227674385418669092818121821, 9.038745585280786078592757515675

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