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Properties

Label 2-2016-7.4-c1-0-1
Degree $2$
Conductor $2016$
Sign $-0.935 - 0.353i$
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  + (−0.342 − 0.593i)5-s + (1.80 + 1.93i)7-s + (−1.76 + 3.05i)11-s − 2.53·13-s + (−2.23 + 3.87i)17-s + (−1.46 − 2.52i)19-s + (1 + 1.73i)23-s + (2.26 − 3.92i)25-s − 6.52·29-s + (0.433 − 0.750i)31-s + (0.531 − 1.73i)35-s + (−1.26 − 2.19i)37-s − 11.6·41-s − 7.39·43-s + (−2.53 − 4.38i)47-s + ⋯
L(s)  = 1  + (−0.153 − 0.265i)5-s + (0.681 + 0.731i)7-s + (−0.532 + 0.922i)11-s − 0.702·13-s + (−0.542 + 0.939i)17-s + (−0.335 − 0.580i)19-s + (0.208 + 0.361i)23-s + (0.453 − 0.784i)25-s − 1.21·29-s + (0.0778 − 0.134i)31-s + (0.0897 − 0.292i)35-s + (−0.208 − 0.360i)37-s − 1.82·41-s − 1.12·43-s + (−0.369 − 0.639i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.935 - 0.353i$
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: \((2,\ 2016,\ (\ :1/2),\ -0.935 - 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5280492080\)
\(L(\frac12)\) \(\approx\) \(0.5280492080\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 1.93i)T \)
good5 \( 1 + (0.342 + 0.593i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.76 - 3.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.53T + 13T^{2} \)
17 \( 1 + (2.23 - 3.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.46 + 2.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.52T + 29T^{2} \)
31 \( 1 + (-0.433 + 0.750i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.26 + 2.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 7.39T + 43T^{2} \)
47 \( 1 + (2.53 + 4.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.12 - 7.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.76 + 8.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.53 - 4.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.82 + 4.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.06T + 71T^{2} \)
73 \( 1 + (-1.26 + 2.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.09 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.53T + 83T^{2} \)
89 \( 1 + (2.05 + 3.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.53T + 97T^{2} \)
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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

−9.452863171490588679521803770252, −8.598104334166079464296806315240, −8.083092839892710559900274445089, −7.18599590120927683537860311412, −6.40623960103164455397230167384, −5.19723110367682868027411158413, −4.88627314653863865503377661745, −3.81873097474905170882839269418, −2.46555518251041896985139407049, −1.77107716770182797284447918480, 0.17541023076022122527814928693, 1.67504277183433012611701493647, 2.90920489733839715456695536755, 3.76857067811178888994515354793, 4.87247489080335147424109914220, 5.38776686186477411063249165691, 6.66626897078195272295347227223, 7.20305392162091897520271294159, 8.066019789364111837417294875017, 8.624025227638394486689681992842

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