Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 7 $
Sign $-0.707 - 0.707i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.86·5-s − 2.64i·7-s − 13.1i·11-s − 5.86·13-s + 0.570·17-s − 15.6i·19-s − 16.4i·23-s + 72.3·25-s + 29.7·29-s − 54.8i·31-s + 26.1i·35-s − 42.0·37-s − 0.773·41-s + 41.7i·43-s − 58.4i·47-s + ⋯
L(s)  = 1  − 1.97·5-s − 0.377i·7-s − 1.19i·11-s − 0.451·13-s + 0.0335·17-s − 0.824i·19-s − 0.716i·23-s + 2.89·25-s + 1.02·29-s − 1.76i·31-s + 0.745i·35-s − 1.13·37-s − 0.0188·41-s + 0.971i·43-s − 1.24i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(2\)
character  :  $\chi_{2016} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2016,\ (\ :1),\ -0.707 - 0.707i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.2290112121\)
\(L(\frac12)\)  \(\approx\)  \(0.2290112121\)
\(L(2)\)   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 \( 1 + 9.86T + 25T^{2} \)
11 \( 1 + 13.1iT - 121T^{2} \)
13 \( 1 + 5.86T + 169T^{2} \)
17 \( 1 - 0.570T + 289T^{2} \)
19 \( 1 + 15.6iT - 361T^{2} \)
23 \( 1 + 16.4iT - 529T^{2} \)
29 \( 1 - 29.7T + 841T^{2} \)
31 \( 1 + 54.8iT - 961T^{2} \)
37 \( 1 + 42.0T + 1.36e3T^{2} \)
41 \( 1 + 0.773T + 1.68e3T^{2} \)
43 \( 1 - 41.7iT - 1.84e3T^{2} \)
47 \( 1 + 58.4iT - 2.20e3T^{2} \)
53 \( 1 + 5.65T + 2.80e3T^{2} \)
59 \( 1 + 42.6iT - 3.48e3T^{2} \)
61 \( 1 + 95.9T + 3.72e3T^{2} \)
67 \( 1 - 69.8iT - 4.48e3T^{2} \)
71 \( 1 - 92.0iT - 5.04e3T^{2} \)
73 \( 1 - 9.97T + 5.32e3T^{2} \)
79 \( 1 + 20.1iT - 6.24e3T^{2} \)
83 \( 1 + 151. iT - 6.88e3T^{2} \)
89 \( 1 + 5.79T + 7.92e3T^{2} \)
97 \( 1 - 103.T + 9.40e3T^{2} \)
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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

−8.398309307562416587100945597024, −7.75222955173482174891921799962, −7.08081646015066529698218112188, −6.27370267159365344976224437091, −4.98836440730888110615512403855, −4.30175850719558041254623781510, −3.51623264768618148489714139785, −2.73026124910802363250251780844, −0.792504659276148726884460527571, −0.085709610164939527483237803853, 1.46751387908249705463144284599, 2.91463483658251957255966941653, 3.70789804139042745582462458317, 4.57981165354464053442934879732, 5.16632234423602671055992528357, 6.57789677644600111727790138427, 7.27754130871114393707060132226, 7.82061664323433935156673047472, 8.540149343127648743228639581536, 9.310494679815914206329071743149

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