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Properties

Label 2-1008-252.31-c1-0-1
Degree $2$
Conductor $1008$
Sign $-0.998 - 0.0605i$
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  + (0.624 + 1.61i)3-s + 2.39i·5-s + (−2.35 + 1.20i)7-s + (−2.21 + 2.01i)9-s + 1.17i·11-s + (3.94 + 2.27i)13-s + (−3.87 + 1.49i)15-s + (−3.59 − 2.07i)17-s + (−0.422 − 0.731i)19-s + (−3.41 − 3.05i)21-s − 3.01i·23-s − 0.741·25-s + (−4.64 − 2.32i)27-s + (1.38 + 2.39i)29-s + (−3.47 − 6.02i)31-s + ⋯
L(s)  = 1  + (0.360 + 0.932i)3-s + 1.07i·5-s + (−0.891 + 0.453i)7-s + (−0.739 + 0.672i)9-s + 0.355i·11-s + (1.09 + 0.631i)13-s + (−0.999 + 0.386i)15-s + (−0.873 − 0.504i)17-s + (−0.0969 − 0.167i)19-s + (−0.744 − 0.667i)21-s − 0.629i·23-s − 0.148·25-s + (−0.894 − 0.447i)27-s + (0.257 + 0.445i)29-s + (−0.624 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.998 - 0.0605i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.998 - 0.0605i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.203655242\)
\(L(\frac12)\) \(\approx\) \(1.203655242\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.624 - 1.61i)T \)
7 \( 1 + (2.35 - 1.20i)T \)
good5 \( 1 - 2.39iT - 5T^{2} \)
11 \( 1 - 1.17iT - 11T^{2} \)
13 \( 1 + (-3.94 - 2.27i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.59 + 2.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.422 + 0.731i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.01iT - 23T^{2} \)
29 \( 1 + (-1.38 - 2.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.47 + 6.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.62 - 8.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.48 + 1.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.19 - 4.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.38 - 9.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.88 + 8.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.78 + 4.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.597 + 0.344i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.56 + 2.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.65iT - 71T^{2} \)
73 \( 1 + (-6.02 - 3.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.7 - 6.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.52 + 4.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.06 - 3.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.316 - 0.182i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.24989951602348602494482972445, −9.590663860627831820006639887177, −8.910152293422615057889907803597, −8.048708914770660472681023540080, −6.67761587345932015672540653537, −6.39267250398193382161384988544, −5.06546701492422534302021734775, −4.01169498841227954852820354258, −3.14223354930308785209933888803, −2.32554751800006882538777614386, 0.51577510290476992184581099476, 1.66762527660223016379626883657, 3.17540470977092875595138574142, 4.00016419358895590320290876735, 5.42449944604947786990018172735, 6.21155034084264211873745511496, 7.00485940969888044829062434751, 8.033496917463712889090253823208, 8.702066059208637902309051623796, 9.228595020546766215700204107277

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