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Properties

Label 2-2016-56.13-c0-0-0
Degree $2$
Conductor $2016$
Sign $-i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 2i·11-s − 25-s + 2i·29-s + 49-s + 2i·53-s − 2i·77-s + 2·79-s − 2i·107-s + ⋯
L(s)  = 1  − 7-s + 2i·11-s − 25-s + 2i·29-s + 49-s + 2i·53-s − 2i·77-s + 2·79-s − 2i·107-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

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8253296309\)
\(L(\frac12)\) \(\approx\) \(0.8253296309\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + T^{2} \)
11 \( 1 - 2iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.495310512200955285690241385389, −9.014765985635509530895120500123, −7.80306595517540421809179504285, −7.14006411746874194488288859304, −6.55357011788315050274679739283, −5.53941744994314085129589988784, −4.63944256059288244050083158848, −3.79327996602954626710338760874, −2.73611361457109315837454974343, −1.66556939907532007794522363574, 0.58263347340642125748447033325, 2.35225331424319618699330017654, 3.36931326510326565572313214757, 3.95556161752243989093016531374, 5.33046400505539845807778232187, 6.09360191673279763545066861955, 6.52773866848835795180167618863, 7.74788973017738012710123004685, 8.337619330154305861744496684188, 9.184846187043044996609270817019

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