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Properties

Label 2-2016-7.4-c1-0-2
Degree $2$
Conductor $2016$
Sign $-0.0627 - 0.998i$
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  + (−1.60 − 2.77i)5-s + (1.02 − 2.43i)7-s + (−2.12 + 3.68i)11-s − 3.15·13-s + (−2.20 + 3.81i)17-s + (1.57 + 2.73i)19-s + (−2.20 − 3.81i)23-s + (−2.62 + 4.54i)25-s − 7.20·29-s + (1.02 − 1.77i)31-s + (−8.40 + 1.06i)35-s + (4.82 + 8.35i)37-s + 10.4·41-s − 0.750·43-s + (1.20 + 2.08i)47-s + ⋯
L(s)  = 1  + (−0.715 − 1.23i)5-s + (0.387 − 0.922i)7-s + (−0.640 + 1.10i)11-s − 0.874·13-s + (−0.533 + 0.924i)17-s + (0.361 + 0.626i)19-s + (−0.459 − 0.795i)23-s + (−0.524 + 0.909i)25-s − 1.33·29-s + (0.183 − 0.318i)31-s + (−1.42 + 0.180i)35-s + (0.793 + 1.37i)37-s + 1.63·41-s − 0.114·43-s + (0.175 + 0.303i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.0627 - 0.998i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.0627 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4703050164\)
\(L(\frac12)\) \(\approx\) \(0.4703050164\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 + 2.43i)T \)
good5 \( 1 + (1.60 + 2.77i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.12 - 3.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.15T + 13T^{2} \)
17 \( 1 + (2.20 - 3.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.57 - 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.20 + 3.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.20T + 29T^{2} \)
31 \( 1 + (-1.02 + 1.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.82 - 8.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 0.750T + 43T^{2} \)
47 \( 1 + (-1.20 - 2.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.64 - 2.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.12 - 7.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.04 - 7.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.57 + 4.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.40T + 71T^{2} \)
73 \( 1 + (7.62 - 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.22 + 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + (-1.20 - 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−9.367641917032093925155793660624, −8.409016698036941445886031786065, −7.72524267497233570147732168730, −7.39739234077632379070800083885, −6.15796283474410126341925196037, −5.05097160679363549016898403524, −4.43991440117219630056343807179, −3.95189276788748560077283383409, −2.35169671312759799069806722249, −1.19830354541306160240875287771, 0.17682678000027265359468528391, 2.31190695290768518627413172043, 2.88650981676102058018954180813, 3.81944258249612148177721423255, 5.04653300330250447935068583154, 5.71015440908578022739075350467, 6.66814425730082856860868795993, 7.58609245794673235087246367398, 7.86486442866553056859019834430, 9.052642039372043176652044217441

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