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Properties

Label 2-2016-3.2-c2-0-15
Degree $2$
Conductor $2016$
Sign $0.816 - 0.577i$
Analytic cond. $54.9320$
Root an. cond. $7.41161$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.54i·5-s − 2.64·7-s + 5.28i·11-s − 17.4·13-s − 7.16i·17-s − 12.1·19-s − 41.2i·23-s + 22.6·25-s − 38.3i·29-s + 45.2·31-s − 4.08i·35-s + 52.5·37-s + 64.3i·41-s − 40.6·43-s + 79.0i·47-s + ⋯
L(s)  = 1  + 0.308i·5-s − 0.377·7-s + 0.480i·11-s − 1.34·13-s − 0.421i·17-s − 0.639·19-s − 1.79i·23-s + 0.904·25-s − 1.32i·29-s + 1.45·31-s − 0.116i·35-s + 1.42·37-s + 1.56i·41-s − 0.946·43-s + 1.68i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(54.9320\)
Root analytic conductor: \(7.41161\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.524121725\)
\(L(\frac12)\) \(\approx\) \(1.524121725\)
\(L(2)\) 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 \( 1 + 2.64T \)
good5 \( 1 - 1.54iT - 25T^{2} \)
11 \( 1 - 5.28iT - 121T^{2} \)
13 \( 1 + 17.4T + 169T^{2} \)
17 \( 1 + 7.16iT - 289T^{2} \)
19 \( 1 + 12.1T + 361T^{2} \)
23 \( 1 + 41.2iT - 529T^{2} \)
29 \( 1 + 38.3iT - 841T^{2} \)
31 \( 1 - 45.2T + 961T^{2} \)
37 \( 1 - 52.5T + 1.36e3T^{2} \)
41 \( 1 - 64.3iT - 1.68e3T^{2} \)
43 \( 1 + 40.6T + 1.84e3T^{2} \)
47 \( 1 - 79.0iT - 2.20e3T^{2} \)
53 \( 1 - 88.4iT - 2.80e3T^{2} \)
59 \( 1 - 39.6iT - 3.48e3T^{2} \)
61 \( 1 - 94.2T + 3.72e3T^{2} \)
67 \( 1 - 13.2T + 4.48e3T^{2} \)
71 \( 1 + 62.1iT - 5.04e3T^{2} \)
73 \( 1 - 12.8T + 5.32e3T^{2} \)
79 \( 1 + 114.T + 6.24e3T^{2} \)
83 \( 1 + 42.7iT - 6.88e3T^{2} \)
89 \( 1 - 9.41iT - 7.92e3T^{2} \)
97 \( 1 - 63.2T + 9.40e3T^{2} \)
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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

−9.129536105164319954544955157455, −8.180776504753416773838917235340, −7.49359384832990905564417283352, −6.58916321520302672909750896851, −6.13399303415147602054707145544, −4.63507320927407006899864295168, −4.50016979600918417383339522837, −2.82077371412249003686532207025, −2.46362206441931871017889786814, −0.75151827712535062268848643320, 0.55146308915181214438918341621, 1.90629244390939533013597411059, 3.00370843209478629228440639749, 3.89516036873953011781525731260, 4.98593216413686857890098259856, 5.55132605725972885774073367484, 6.67353258902195681793429727359, 7.21362179364729734673804773016, 8.261098756210974992716573179118, 8.794320236524023190030567169158

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