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Properties

Label 2-2016-504.349-c0-0-0
Degree $2$
Conductor $2016$
Sign $-0.766 - 0.642i$
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  + i·3-s + (0.866 + 1.5i)5-s + (−0.5 + 0.866i)7-s − 9-s + (−1.5 + 0.866i)15-s + 1.73·19-s + (−0.866 − 0.5i)21-s + (−0.5 − 0.866i)23-s + (−1 + 1.73i)25-s − i·27-s − 1.73·35-s + (−0.866 − 1.5i)45-s + (−0.499 − 0.866i)49-s + 1.73i·57-s + (0.866 − 1.5i)61-s + ⋯
L(s)  = 1  + i·3-s + (0.866 + 1.5i)5-s + (−0.5 + 0.866i)7-s − 9-s + (−1.5 + 0.866i)15-s + 1.73·19-s + (−0.866 − 0.5i)21-s + (−0.5 − 0.866i)23-s + (−1 + 1.73i)25-s − i·27-s − 1.73·35-s + (−0.866 − 1.5i)45-s + (−0.499 − 0.866i)49-s + 1.73i·57-s + (0.866 − 1.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216248040\)
\(L(\frac12)\) \(\approx\) \(1.216248040\)
\(L(1)\) 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 \( 1 - iT \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.720958693239544516412991088415, −9.202366824180980329185398876704, −8.207204606037178799322616889381, −7.14629185954236634637976899980, −6.29071246239037714881285527087, −5.75725510399000693920710996602, −4.96701549382516407527698778704, −3.59039257351819245297402078647, −2.97544381771603144949998445985, −2.21945229255054561772703182570, 0.914999951751462959605172418322, 1.66511766581333549435910637309, 2.99016122737909091099703452434, 4.15707548533861419226575951639, 5.31945998858142064469763941917, 5.71962191445112787303397919292, 6.72558198284030898067864795085, 7.51761926531715047673977397018, 8.178139038710266661924761978685, 9.110909029997048249332493909666

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