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Properties

Label 2-2016-7.4-c1-0-29
Degree $2$
Conductor $2016$
Sign $-0.701 + 0.712i$
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 + 0.866i)5-s + (−2.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−4 + 6.92i)17-s + (−2 − 3.46i)19-s + (−2 − 3.46i)23-s + (2 − 3.46i)25-s + 5·29-s + (3.5 − 6.06i)31-s + (−2 − 1.73i)35-s + (−4 − 6.92i)37-s − 4·41-s − 10·43-s + (−3 − 5.19i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.944 + 0.327i)7-s + (−0.150 + 0.261i)11-s + (−0.970 + 1.68i)17-s + (−0.458 − 0.794i)19-s + (−0.417 − 0.722i)23-s + (0.400 − 0.692i)25-s + 0.928·29-s + (0.628 − 1.08i)31-s + (−0.338 − 0.292i)35-s + (−0.657 − 1.13i)37-s − 0.624·41-s − 1.52·43-s + (−0.437 − 0.757i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.701 + 0.712i$
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: \((2,\ 2016,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2827921097\)
\(L(\frac12)\) \(\approx\) \(0.2827921097\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (4 - 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−8.671402040039171612653518383058, −8.381268368829245021034740366275, −7.03510195957839761696776047747, −6.47055911343559578408169538112, −5.94290484005618604449461044539, −4.70395015708823474250846296523, −3.89606790659424376567278460648, −2.78166242170178377491340111967, −2.01671308047176001078828799269, −0.098224043302472275112384485964, 1.36738530905757923918288179046, 2.77860272327023968410816675141, 3.52202850614738025942045694394, 4.71207873875608928636759104508, 5.32828543235199661118540000136, 6.56574450390117262215374100616, 6.80454078301879247110257924812, 7.980503811891250433713571878306, 8.706945779487115974523134438209, 9.522521699464241766746872897672

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