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Properties

Label 2-2016-7.4-c1-0-12
Degree $2$
Conductor $2016$
Sign $0.991 - 0.126i$
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  + (−2.5 − 0.866i)7-s + (−1 + 1.73i)11-s + 13-s + (−1 + 1.73i)17-s + (−2.5 − 4.33i)19-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + 8·29-s + (1.5 − 2.59i)31-s + (4.5 + 7.79i)37-s − 2·41-s + 43-s + (4 + 6.92i)47-s + (5.5 + 4.33i)49-s + (3 − 5.19i)53-s + ⋯
L(s)  = 1  + (−0.944 − 0.327i)7-s + (−0.301 + 0.522i)11-s + 0.277·13-s + (−0.242 + 0.420i)17-s + (−0.573 − 0.993i)19-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + 1.48·29-s + (0.269 − 0.466i)31-s + (0.739 + 1.28i)37-s − 0.312·41-s + 0.152·43-s + (0.583 + 1.01i)47-s + (0.785 + 0.618i)49-s + (0.412 − 0.713i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.991 - 0.126i$
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: \((2,\ 2016,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451447220\)
\(L(\frac12)\) \(\approx\) \(1.451447220\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−9.210220126892870990936180394985, −8.428174046964265616624622357564, −7.57565832573131320214356719704, −6.63368033627424095526584596572, −6.29429945650867780502041776543, −5.01781905411925733435518817575, −4.30573812498332139594262724586, −3.23645231162302387639856389423, −2.39035303845600298420578278056, −0.835380808261731764761490791627, 0.75548003203376718681735610263, 2.38941155371295873847834442547, 3.18319680120722497899312602920, 4.14319712929519341650642299714, 5.20647901203443763894440876646, 6.04700827487207487213898640207, 6.67866005004160149304255744705, 7.51874296049311745459624114336, 8.729983498780691958254515901069, 8.802887512635859271236806134655

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