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Properties

Label 2-2016-7.4-c1-0-0
Degree $2$
Conductor $2016$
Sign $-0.922 - 0.386i$
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  + (0.5 + 0.866i)5-s + (−1.73 − 2i)7-s + (−2.59 + 4.5i)11-s + (2.5 − 4.33i)17-s + (0.866 + 1.5i)19-s + (0.866 + 1.5i)23-s + (2 − 3.46i)25-s − 8·29-s + (−4.33 + 7.5i)31-s + (0.866 − 2.5i)35-s + (2.5 + 4.33i)37-s − 4·41-s − 6.92·43-s + (−4.33 − 7.5i)47-s + (−1.00 + 6.92i)49-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.654 − 0.755i)7-s + (−0.783 + 1.35i)11-s + (0.606 − 1.05i)17-s + (0.198 + 0.344i)19-s + (0.180 + 0.312i)23-s + (0.400 − 0.692i)25-s − 1.48·29-s + (−0.777 + 1.34i)31-s + (0.146 − 0.422i)35-s + (0.410 + 0.711i)37-s − 0.624·41-s − 1.05·43-s + (−0.631 − 1.09i)47-s + (−0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.922 - 0.386i$
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: \((2,\ 2016,\ (\ :1/2),\ -0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4093863277\)
\(L(\frac12)\) \(\approx\) \(0.4093863277\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 2i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.59 - 4.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.866 - 1.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.866 - 1.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (4.33 - 7.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 6.92T + 43T^{2} \)
47 \( 1 + (4.33 + 7.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 - 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−9.827495744714156067332465783112, −8.792297001876570668302488848322, −7.62139002029325895329258891195, −7.22799474634990997044360328860, −6.53886626359238069758819959083, −5.39589148667742792058796890741, −4.72758042164038196313092923396, −3.58138971344786329739863231017, −2.80091824956010040611232995517, −1.57820317389206327820091561487, 0.13882210585428058306320824275, 1.72163364649437463951143090564, 2.95597875702242550220827768571, 3.60854335497047465632717484046, 4.93273931025385622770627041957, 5.89024236579650227554949090805, 5.98867641304559880921677687964, 7.37584578015453126851073783160, 8.098086332986839950177482496139, 8.946121690477604725387544546756

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