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Properties

Label 2-2016-504.499-c0-0-0
Degree $2$
Conductor $2016$
Sign $0.971 - 0.235i$
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  + 3-s + (−0.866 + 0.5i)5-s − i·7-s + 9-s + (−0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − i·21-s + (0.866 − 0.5i)23-s + 27-s + (−0.866 + 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯
L(s)  = 1  + 3-s + (−0.866 + 0.5i)5-s − i·7-s + 9-s + (−0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − i·21-s + (0.866 − 0.5i)23-s + 27-s + (−0.866 + 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.497729248\)
\(L(\frac12)\) \(\approx\) \(1.497729248\)
\(L(1)\) 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 \( 1 - T \)
7 \( 1 + iT \)
good5 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.298979959879874296688582488868, −8.499319340976909184680925155318, −7.71032485523505100559368044197, −7.20933111378829073322847311802, −6.64867806659210168536982905695, −5.14163316858762065888393606682, −4.08589529023136171634107614193, −3.70051467709518397146510503186, −2.69039443357385505142770334513, −1.39083920379539830629119355680, 1.21600198531380253140492633867, 2.72685212751734310627748068794, 3.32490906733874873109772551917, 4.23622190833605931192761261269, 5.33687866218894042104795589000, 5.99238593005154809693645393088, 7.31577227740803053383002885185, 8.062788948886037319567431432696, 8.322088411619331535590297798613, 9.217246590838677211255777616815

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