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Properties

Label 2-2016-56.53-c1-0-18
Degree $2$
Conductor $2016$
Sign $0.944 + 0.329i$
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  + (−1.56 + 0.902i)5-s + (−2.63 − 0.217i)7-s + (−4.48 − 2.58i)11-s + 0.840i·13-s + (−2.45 + 4.25i)17-s + (4.87 − 2.81i)19-s + (3.05 + 5.28i)23-s + (−0.872 + 1.51i)25-s − 0.439i·29-s + (3.66 − 6.35i)31-s + (4.31 − 2.03i)35-s + (4.56 − 2.63i)37-s + 6.23·41-s − 7.34i·43-s + (−2.83 − 4.91i)47-s + ⋯
L(s)  = 1  + (−0.698 + 0.403i)5-s + (−0.996 − 0.0820i)7-s + (−1.35 − 0.779i)11-s + 0.232i·13-s + (−0.595 + 1.03i)17-s + (1.11 − 0.646i)19-s + (0.636 + 1.10i)23-s + (−0.174 + 0.302i)25-s − 0.0816i·29-s + (0.658 − 1.14i)31-s + (0.729 − 0.344i)35-s + (0.750 − 0.433i)37-s + 0.974·41-s − 1.12i·43-s + (−0.413 − 0.716i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ 0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9628355367\)
\(L(\frac12)\) \(\approx\) \(0.9628355367\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.217i)T \)
good5 \( 1 + (1.56 - 0.902i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.48 + 2.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.840iT - 13T^{2} \)
17 \( 1 + (2.45 - 4.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.87 + 2.81i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.05 - 5.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.439iT - 29T^{2} \)
31 \( 1 + (-3.66 + 6.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.56 + 2.63i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 + 7.34iT - 43T^{2} \)
47 \( 1 + (2.83 + 4.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.15 - 0.669i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.31 + 4.22i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.77 + 2.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.647 + 0.373i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.08T + 71T^{2} \)
73 \( 1 + (-3.70 + 6.42i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.68 - 15.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.45iT - 83T^{2} \)
89 \( 1 + (-3.10 - 5.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.81T + 97T^{2} \)
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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

−9.223044586864097454138993014638, −8.160120466880479903545850625793, −7.59510434321051968362106188195, −6.83278464270357073142677451253, −5.93570905232611173768601452731, −5.19911136692819664809890184172, −3.93397141523609961426310418601, −3.29416279509661885352814792642, −2.41257041107247033947309585816, −0.53798277873867917655961827067, 0.73137639419325880759795266377, 2.56066362397033143114884649117, 3.16358920622910205488037254709, 4.46815044917388350287662360637, 4.98454643009237927242169869094, 6.04444774397681339384685905792, 6.96262706379756645087635972680, 7.65735816340138737147085560154, 8.309443735393178186533612510317, 9.276655324239516876653548143709

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