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Properties

Label 2-1008-252.115-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.942 - 0.333i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  − 1.73i·3-s + (3.62 + 2.09i)5-s + (1 + 2.44i)7-s − 2.99·9-s + (−1.5 + 0.866i)11-s + (0.621 − 0.358i)13-s + (3.62 − 6.27i)15-s + (5.74 + 3.31i)17-s + (0.5 + 0.866i)19-s + (4.24 − 1.73i)21-s + (−6.62 − 3.82i)23-s + (6.24 + 10.8i)25-s + 5.19i·27-s + (−3.62 + 6.27i)29-s + 4·31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (1.61 + 0.935i)5-s + (0.377 + 0.925i)7-s − 0.999·9-s + (−0.452 + 0.261i)11-s + (0.172 − 0.0994i)13-s + (0.935 − 1.61i)15-s + (1.39 + 0.804i)17-s + (0.114 + 0.198i)19-s + (0.925 − 0.377i)21-s + (−1.38 − 0.797i)23-s + (1.24 + 2.16i)25-s + 0.999i·27-s + (−0.672 + 1.16i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.109284900\)
\(L(\frac12)\) \(\approx\) \(2.109284900\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 + (-3.62 - 2.09i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.621 + 0.358i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.74 - 3.31i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.62 + 3.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.62 - 6.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.257 - 0.148i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.74 + 1.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + (-3.62 + 6.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 6.33iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 11.8iT - 79T^{2} \)
83 \( 1 + (-2.74 + 4.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.2 + 6.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.74 - 1.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.17576945607908359195822907692, −9.141681839403262118325085415211, −8.298521252766716272703270849366, −7.43650563563790192929922512307, −6.46285195571087344142202047503, −5.79135299973560000147760190006, −5.35012750745835212495512374283, −3.28804122939699773957817524262, −2.28781298206815606137789073820, −1.67303549482966478457324910943, 1.03476894378478545090356077866, 2.44911386473045651008386245777, 3.79764254009963765980089672984, 4.80673353384998062447147785189, 5.49818713735776871252522519905, 6.11016725199287614420477759324, 7.60234295603218662908612914851, 8.436865887391115976394540286992, 9.340870506179773064652001142682, 10.08335042889671983281943473330

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