Properties

Label 2-2016-7.4-c1-0-7
Degree $2$
Conductor $2016$
Sign $-0.266 - 0.963i$
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 + 2.59i)7-s + (−1 + 1.73i)11-s + 5·13-s + (−1 + 1.73i)17-s + (1.5 + 2.59i)19-s + (−1 − 1.73i)23-s + (2.5 − 4.33i)25-s − 8·29-s + (−0.5 + 0.866i)31-s + (2.5 + 4.33i)37-s − 2·41-s − 7·43-s + (4 + 6.92i)47-s + (−6.5 − 2.59i)49-s + (−1 + 1.73i)53-s + ⋯
L(s)  = 1  + (−0.188 + 0.981i)7-s + (−0.301 + 0.522i)11-s + 1.38·13-s + (−0.242 + 0.420i)17-s + (0.344 + 0.596i)19-s + (−0.208 − 0.361i)23-s + (0.5 − 0.866i)25-s − 1.48·29-s + (−0.0898 + 0.155i)31-s + (0.410 + 0.711i)37-s − 0.312·41-s − 1.06·43-s + (0.583 + 1.01i)47-s + (−0.928 − 0.371i)49-s + (−0.137 + 0.237i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 2016,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.362297927\)
\(L(\frac12)\) \(\approx\) \(1.362297927\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.5 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.321373833852006787912146243776, −8.534248896308258362089089932393, −8.048392526185724101371082483233, −6.93641609323326693702872909512, −6.10808034512077537490277602006, −5.56248223166213060823120987528, −4.48280947861078819784756692361, −3.54599306599638251290323240384, −2.52824361942447923571916898319, −1.45543627586309498611810570685, 0.49884445031552276713561240693, 1.74243995647585660642647000863, 3.27656771540192437820912824979, 3.75131724882290840631377320753, 4.89757100787011739868456377650, 5.74853529501354851062450096635, 6.62864834393501251245400976408, 7.35562504577457230625684515014, 8.087817976670662742418047574168, 8.999333564923491531527365074893

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