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Properties

Label 2-2016-168.11-c1-0-24
Degree $2$
Conductor $2016$
Sign $-0.988 + 0.153i$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.914 − 1.58i)5-s + (−0.358 + 2.62i)7-s + (1.37 + 0.792i)11-s + 2.58i·13-s + (−5.40 − 3.12i)17-s + (−1.29 − 2.23i)19-s + (0.292 + 0.507i)23-s + (0.828 − 1.43i)25-s − 3·29-s + (3.82 + 2.20i)31-s + (4.47 − 1.82i)35-s + (−7.85 + 4.53i)37-s − 8.82i·41-s − 11.4·43-s + (−1.29 − 2.23i)47-s + ⋯
L(s)  = 1  + (−0.408 − 0.708i)5-s + (−0.135 + 0.990i)7-s + (0.414 + 0.239i)11-s + 0.717i·13-s + (−1.31 − 0.757i)17-s + (−0.296 − 0.513i)19-s + (0.0610 + 0.105i)23-s + (0.165 − 0.286i)25-s − 0.557·29-s + (0.686 + 0.396i)31-s + (0.757 − 0.309i)35-s + (−1.29 + 0.745i)37-s − 1.37i·41-s − 1.74·43-s + (−0.188 − 0.326i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.988 + 0.153i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1054924311\)
\(L(\frac12)\) \(\approx\) \(0.1054924311\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.358 - 2.62i)T \)
good5 \( 1 + (0.914 + 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.37 - 0.792i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.58iT - 13T^{2} \)
17 \( 1 + (5.40 + 3.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.29 + 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.292 - 0.507i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-3.82 - 2.20i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.85 - 4.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.82iT - 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (1.29 + 2.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.08 + 5.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.98 - 4.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.40 + 3.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.82 - 6.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (-3.65 + 6.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.98 - 4.03i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.07iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−8.745595353474107564537616916710, −8.358339324236149508190102841519, −6.94852931418534608498292378343, −6.65746277388608643705086102142, −5.38393215346680521729242096513, −4.76902377219045111413069735784, −3.93484737715813498151994136120, −2.69110680792532251523493695603, −1.71975438447338884102789651369, −0.03654408109649500806828541770, 1.53644752446166609774485316556, 2.92978539549763392679232848476, 3.76007177424683492509706199996, 4.42477657152694838755734899730, 5.63864246470520817104196931037, 6.62707559082757321466790340418, 7.01047271848382251594473898760, 7.997529030097576897315008025359, 8.561576044232899543067742007725, 9.647705841036907043896964676807

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