Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·5-s i·7-s + 1.41·11-s − 1.41i·17-s + 2i·19-s − 1.41·23-s − 1.00·25-s − 1.41·35-s − 1.41i·41-s − 49-s − 2.00i·55-s + 1.41·71-s − 1.41i·77-s − 2.00·85-s + 1.41i·89-s + ⋯
L(s)  = 1  − 1.41i·5-s i·7-s + 1.41·11-s − 1.41i·17-s + 2i·19-s − 1.41·23-s − 1.00·25-s − 1.41·35-s − 1.41i·41-s − 49-s − 2.00i·55-s + 1.41·71-s − 1.41i·77-s − 2.00·85-s + 1.41i·89-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $i$
motivic weight  =  \(0\)
character  :  $\chi_{2016} (1441, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2016,\ (\ :0),\ i)$
$L(\frac{1}{2})$  $\approx$  $1.183122469$
$L(\frac12)$  $\approx$  $1.183122469$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 1.41iT - T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.243428147065903774952406887985, −8.338299410909366001810149279223, −7.72795850514979939633403928328, −6.81602805952662671527989733065, −5.91417226871951285500437089425, −5.04820317817149915537790032745, −4.10731287785860528203320251107, −3.69769869507050417664612806438, −1.85049143501637111046529227519, −0.911336349285537840221428926381, 1.83053504815339100853283845117, 2.75275375171096306086207964270, 3.62851429915875349752060224539, 4.57235103219738370142888263103, 5.87805167280411652657719349527, 6.41620939846779624137200615489, 6.95067350979800952610770344633, 8.024783850143527977247641519413, 8.796876090294785283832789724259, 9.525831406850213719209529330084

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