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Properties

Label 2-2016-672.83-c0-0-3
Degree $2$
Conductor $2016$
Sign $-0.933 - 0.358i$
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·2-s − 4-s + (−0.707 − 0.707i)7-s + i·8-s + (−0.292 − 0.707i)11-s + (−0.707 + 0.707i)14-s + 16-s + (−0.707 + 0.292i)22-s + (−1.41 − 1.41i)23-s + (−0.707 + 0.707i)25-s + (0.707 + 0.707i)28-s + (−1.70 − 0.707i)29-s − i·32-s + (1.70 − 0.707i)37-s + (−0.707 + 0.292i)43-s + (0.292 + 0.707i)44-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.707 − 0.707i)7-s + i·8-s + (−0.292 − 0.707i)11-s + (−0.707 + 0.707i)14-s + 16-s + (−0.707 + 0.292i)22-s + (−1.41 − 1.41i)23-s + (−0.707 + 0.707i)25-s + (0.707 + 0.707i)28-s + (−1.70 − 0.707i)29-s − i·32-s + (1.70 − 0.707i)37-s + (−0.707 + 0.292i)43-s + (0.292 + 0.707i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.933 - 0.358i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ -0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4983550888\)
\(L(\frac12)\) \(\approx\) \(0.4983550888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
29 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
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

−9.330095720797944351405316146445, −8.082389523709181232238611133131, −7.69237710651917769225235203678, −6.31660536756431502932606478048, −5.72902113038716905034427080292, −4.49475995485367125984307323852, −3.82672714454300846548913774749, −2.99054064300902038848211216543, −1.90083434302803730574946906646, −0.34391161518839371166495407209, 1.93035341630232667844571464797, 3.31467915592426290155433760809, 4.17854997823022952060574880953, 5.22029618327676392781581150953, 5.89207665953990287976433038248, 6.54137597134582445942489343045, 7.52956495152331844457543677564, 8.003923626208591220136700511421, 8.975322268796624956607057208605, 9.763372458773832981002184429214

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