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Properties

Label 2-2016-672.587-c0-0-2
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  + 2-s + 4-s + (−0.707 + 0.707i)7-s + 8-s + (0.707 + 0.292i)11-s + (−0.707 + 0.707i)14-s + 16-s + (0.707 + 0.292i)22-s + (0.707 + 0.707i)25-s + (−0.707 + 0.707i)28-s + (−0.707 − 1.70i)29-s + 32-s + (−0.292 + 0.707i)37-s + (−0.707 + 1.70i)43-s + (0.707 + 0.292i)44-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−0.707 + 0.707i)7-s + 8-s + (0.707 + 0.292i)11-s + (−0.707 + 0.707i)14-s + 16-s + (0.707 + 0.292i)22-s + (0.707 + 0.707i)25-s + (−0.707 + 0.707i)28-s + (−0.707 − 1.70i)29-s + 32-s + (−0.292 + 0.707i)37-s + (−0.707 + 1.70i)43-s + (0.707 + 0.292i)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: $\chi_{2016} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.158270120\)
\(L(\frac12)\) \(\approx\) \(2.158270120\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good5 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
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

−9.536258721331738727696632878067, −8.573656742177748377868043290133, −7.63949053865133953819412636171, −6.73286939350955501460688611893, −6.21090905980475640858097136447, −5.39104287709016801832442784285, −4.50485158055025781113533528055, −3.57928445123212443906186708855, −2.77389108253624107216387536119, −1.68764139334497581913424624736, 1.34491021891440770966109071727, 2.73522286090890548931560528823, 3.62238035867631709600739697164, 4.23431157981915165963925060638, 5.29542797413871424001298801107, 6.07447399905347671047239071367, 6.97467575411645836447957178940, 7.25224170970238946719627496066, 8.523245711491623987210102476597, 9.284440489113700431419774130126

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