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Properties

Label 2-2016-56.5-c0-0-0
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

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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

\(L(\frac{1}{2})\) \(\approx\) \(0.7451390209\)
\(L(\frac12)\) \(\approx\) \(0.7451390209\)
\(L(1)\) 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 \( 1 - 1.73iT - T^{2} \)
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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

−9.714779237536054059240222403345, −8.771850701321860627250103152000, −7.85708040378995423149297443617, −7.46849838338071215596242800391, −6.60319234581935112019081635738, −5.72537615874244332257885355216, −4.80731484587997741225077686461, −3.76390333179763627730854732260, −2.85259609683907292955774033930, −2.10298852297526176634389596468, 0.52528477180275316985364138298, 1.75349527851813481850706749980, 3.40658408062262507483547913659, 4.15623370320562464879465113375, 5.00001713093185596057047142768, 5.49801486711229229540776655991, 6.86583713712674921283527437098, 7.79664977162585715740132016618, 8.094417332004570304552561150505, 8.905802487019823531987217588136

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