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Properties

Label 2-2016-7.2-c1-0-22
Degree $2$
Conductor $2016$
Sign $0.905 + 0.424i$
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  + (1.37 − 2.37i)5-s + (−2.64 + 0.0585i)7-s + (0.771 + 1.33i)11-s + 6.03·13-s + (3.74 + 6.48i)17-s + (3.01 − 5.22i)19-s + (−3.74 + 6.48i)23-s + (−1.27 − 2.20i)25-s − 1.25·29-s + (−2.64 − 4.58i)31-s + (−3.49 + 6.37i)35-s + (−2.47 + 4.28i)37-s + 5.08·41-s + 3.45·43-s + (4.74 − 8.22i)47-s + ⋯
L(s)  = 1  + (0.614 − 1.06i)5-s + (−0.999 + 0.0221i)7-s + (0.232 + 0.403i)11-s + 1.67·13-s + (0.908 + 1.57i)17-s + (0.692 − 1.19i)19-s + (−0.781 + 1.35i)23-s + (−0.254 − 0.440i)25-s − 0.232·29-s + (−0.475 − 0.822i)31-s + (−0.590 + 1.07i)35-s + (−0.406 + 0.704i)37-s + 0.794·41-s + 0.527·43-s + (0.692 − 1.19i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.905 + 0.424i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ 0.905 + 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.984288988\)
\(L(\frac12)\) \(\approx\) \(1.984288988\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0585i)T \)
good5 \( 1 + (-1.37 + 2.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.771 - 1.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.03T + 13T^{2} \)
17 \( 1 + (-3.74 - 6.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.01 + 5.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.74 - 6.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.25T + 29T^{2} \)
31 \( 1 + (2.64 + 4.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.47 - 4.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.08T + 41T^{2} \)
43 \( 1 - 3.45T + 43T^{2} \)
47 \( 1 + (-4.74 + 8.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.91 + 3.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.77 - 4.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.29 + 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.01 - 3.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 + (6.27 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.89 + 6.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.52T + 83T^{2} \)
89 \( 1 + (4.74 - 8.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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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

−9.203640752353504766768482220385, −8.475261798092388375794953347776, −7.62423097883120224862282884987, −6.54101782481196936698632620648, −5.84502121091318151723033068212, −5.31960199607841318550227457438, −3.97725086844960362205797236678, −3.45789982026229157690074234889, −1.90786065007218142949134271486, −0.968880721461103888182901828628, 1.01035654160233090890487387945, 2.53533863204684771207572698904, 3.28532768333680895484636049297, 3.98798079304625088676741011186, 5.62005292375709409612941816938, 6.00094939684712578043807848080, 6.75914831247798407906702242912, 7.48288128276373269245756238292, 8.511156354808416042246153933782, 9.333878015571731984451298189166

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