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Properties

Label 2-2016-672.419-c0-0-0
Degree $2$
Conductor $2016$
Sign $0.999 + 0.0264i$
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 + (−1.70 + 0.707i)11-s + (0.707 − 0.707i)14-s + 16-s + (0.707 + 1.70i)22-s + (1.41 + 1.41i)23-s + (0.707 − 0.707i)25-s + (−0.707 − 0.707i)28-s + (−0.292 + 0.707i)29-s − i·32-s + (0.292 + 0.707i)37-s + (0.707 + 1.70i)43-s + (1.70 − 0.707i)44-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.707 + 0.707i)7-s + i·8-s + (−1.70 + 0.707i)11-s + (0.707 − 0.707i)14-s + 16-s + (0.707 + 1.70i)22-s + (1.41 + 1.41i)23-s + (0.707 − 0.707i)25-s + (−0.707 − 0.707i)28-s + (−0.292 + 0.707i)29-s − i·32-s + (0.292 + 0.707i)37-s + (0.707 + 1.70i)43-s + (1.70 − 0.707i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.999 + 0.0264i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.999 + 0.0264i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9606626638\)
\(L(\frac12)\) \(\approx\) \(0.9606626638\)
\(L(1)\) 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 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
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

−9.470275126831980317184789936288, −8.650825968398900025396290888186, −7.973610050013904026123294759884, −7.26203509218832105987449553217, −5.81026686113861551141785463727, −5.02456842838157491534740933452, −4.64232388546465081000272196256, −3.16485132385389032180287234293, −2.52728407753931018871541554280, −1.45534177058383156583951114427, 0.75272173311407154287499577779, 2.60494098744945403298028760799, 3.75981324920675879534514798372, 4.80327268371171929098204209343, 5.26900824355964472708223216374, 6.18259164098407931160302873731, 7.24802226714472759799218819752, 7.60578026009741725032949435181, 8.495087410482806839553270956017, 8.959203899840502667662407527684

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