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Properties

Label 2-2016-7.2-c1-0-14
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: $\chi_{2016} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ 0.935 - 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.510233270\)
\(L(\frac12)\) \(\approx\) \(1.510233270\)
\(L(\frac{3}{2})\) 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

−9.150907265222098195558349229192, −8.523672028408805239024037043772, −7.71701153345561800537567231166, −6.77489385259806570345971335436, −5.87877357166352715428620132108, −5.38500854898667544159083188322, −4.33040821984106440896129820037, −3.11697296071943744356433749320, −2.51685942091837467605338740616, −0.914720478277430616235403199149, 0.72392837991275666731553679719, 2.32037362503896250398253788002, 3.12580041112061492478075629516, 4.22716956178800274429457310742, 5.02696551375294500791890041807, 5.95508936235480753271283620323, 7.03294493573913052167983633369, 7.29662701009928358262126521838, 8.204961722953856812032654852270, 9.353864831666851668245152919586

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