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Properties

Label 2-2016-672.83-c0-0-1
Degree $2$
Conductor $2016$
Sign $0.0264 - 0.999i$
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

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

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.0264 - 0.999i$
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: \((2,\ 2016,\ (\ :0),\ 0.0264 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9665183247\)
\(L(\frac12)\) \(\approx\) \(0.9665183247\)
\(L(1)\) 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

−9.472368788713987986212462950841, −8.755340271323984309367842669399, −7.69615289939385086964684656108, −7.17262870861919062751944519403, −6.55985190052859859002576139531, −5.65079895785379021835174337230, −4.76986827466929737151165882608, −3.94894410882725280144972409424, −3.05906925118859101995949281741, −1.18996259349701277342310735832, 0.872598133267130174409005812755, 2.49679569213102879177197139766, 2.97564117624893100135275003344, 4.11231479139881535763975438606, 4.90964659319517410040358161467, 5.98333640734353511438131131760, 6.54251017991441677279020972744, 7.908463517289669584470219308247, 8.703526388435529396559950884078, 9.091665088729017736851864948080

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