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Properties

Label 2-2016-28.23-c0-0-1
Degree $2$
Conductor $2016$
Sign $0.319 + 0.947i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $0$
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 − i·7-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + (−0.866 + 0.5i)31-s + (−0.866 − 0.5i)35-s + (0.5 − 0.866i)37-s + (0.866 + 0.5i)47-s − 49-s + (−0.5 − 0.866i)53-s − 0.999i·55-s + (−0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s − i·7-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + (−0.866 + 0.5i)31-s + (−0.866 − 0.5i)35-s + (0.5 − 0.866i)37-s + (0.866 + 0.5i)47-s − 49-s + (−0.5 − 0.866i)53-s − 0.999i·55-s + (−0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.319 + 0.947i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.319 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273044162\)
\(L(\frac12)\) \(\approx\) \(1.273044162\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.163839659225115629027129092302, −8.481577428880311606551780774340, −7.73173320989464983856948393597, −6.71098384564245410073796613728, −6.07750704687062031585181274565, −5.14426014108578891396799577458, −4.19756026004570153111352147272, −3.58886916400583979528153467484, −2.01786355941077241713527930167, −0.977336590523553512391287653304, 1.78806040064603763143423482924, 2.58832097175427601810190195703, 3.59826379470201509323266010958, 4.65752590781038788970274637330, 5.75563409448769321394720435853, 6.24111988969012474158524434410, 7.06489503333740383019877768371, 7.905326207212580453424567740711, 8.846501447174143898854959441623, 9.553600870431373685521392666129

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