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Properties

Label 2-2016-7.4-c1-0-10
Degree $2$
Conductor $2016$
Sign $-0.605 - 0.795i$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)5-s + (0.5 + 2.59i)7-s + (−2.5 + 4.33i)11-s + 2·13-s + (1 − 1.73i)17-s + (3 + 5.19i)19-s + (−1 − 1.73i)23-s + (−2 + 3.46i)25-s − 29-s + (0.5 − 0.866i)31-s + (−6 + 5.19i)35-s + (−5 − 8.66i)37-s − 4·41-s + 4·43-s + (4 + 6.92i)47-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)5-s + (0.188 + 0.981i)7-s + (−0.753 + 1.30i)11-s + 0.554·13-s + (0.242 − 0.420i)17-s + (0.688 + 1.19i)19-s + (−0.208 − 0.361i)23-s + (−0.400 + 0.692i)25-s − 0.185·29-s + (0.0898 − 0.155i)31-s + (−1.01 + 0.878i)35-s + (−0.821 − 1.42i)37-s − 0.624·41-s + 0.609·43-s + (0.583 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.798490892\)
\(L(\frac12)\) \(\approx\) \(1.798490892\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.5 + 4.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.5 - 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 19T + 97T^{2} \)
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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

−9.525774627127339485037275891585, −8.736691780547568924835719545975, −7.65793701233622131700668379039, −7.20361048881376163126012316129, −6.07935480851325592963968882750, −5.67423549159683224908794303725, −4.66948933101685461394439327788, −3.40943219604235664577552860876, −2.50033588674012115380288386634, −1.80777817784617706108728586105, 0.64836441377961573383700110765, 1.51261375670747518773645512088, 2.99938453640640460508747082895, 3.93798226426328505923379919355, 5.03700057765526876492813317456, 5.46896932051028645702667004199, 6.43638663487877139478220909891, 7.38534481682280194667432039612, 8.346747057119739796582791129875, 8.685204516555070348500925097758

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