↑

Properties

Label 2-2016-56.5-c0-0-1
Degree $2$
Conductor $2016$
Sign $0.795 + 0.605i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)5-s + (0.5 + 0.866i)7-s + (0.866 − 0.5i)11-s + (−1 − 1.73i)25-s + i·29-s + (−1.5 + 0.866i)31-s + 1.73·35-s + (−0.499 + 0.866i)49-s + (0.866 − 0.5i)53-s − 1.73i·55-s + (−0.866 − 1.5i)59-s + (0.866 + 0.499i)77-s + (0.5 − 0.866i)79-s − 1.73·83-s + 1.73i·97-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)5-s + (0.5 + 0.866i)7-s + (0.866 − 0.5i)11-s + (−1 − 1.73i)25-s + i·29-s + (−1.5 + 0.866i)31-s + 1.73·35-s + (−0.499 + 0.866i)49-s + (0.866 − 0.5i)53-s − 1.73i·55-s + (−0.866 − 1.5i)59-s + (0.866 + 0.499i)77-s + (0.5 − 0.866i)79-s − 1.73·83-s + 1.73i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.795 + 0.605i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.795 + 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.474528847\)
\(L(\frac12)\) \(\approx\) \(1.474528847\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.968449389284427238641763482313, −8.837947718689458514013551694901, −7.970918842062309682411828197433, −6.77017211782634985405107076379, −5.84325780958923743493269338971, −5.30257849445462530020878861480, −4.62312554575023135357416503864, −3.45555184705016807768510337057, −2.04175386152273353527504027886, −1.28797238410692487110327069689, 1.59517689284999459059657863235, 2.50744296080747164808441305408, 3.63838132946721242100265009044, 4.37773044823573934479972895224, 5.66040613165431034923401088686, 6.31254023219666110765394716893, 7.21644970861672826768641830720, 7.45292848686517787503435491901, 8.721864357359744306425747031840, 9.720107363261306515883837085759

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