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Properties

Label 2-1008-63.16-c1-0-6
Degree $2$
Conductor $1008$
Sign $-0.924 - 0.381i$
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  + (−0.371 + 1.69i)3-s + 1.68·5-s + (−0.960 + 2.46i)7-s + (−2.72 − 1.25i)9-s − 1.24·11-s + (1.96 + 3.39i)13-s + (−0.626 + 2.84i)15-s + (−1.62 − 2.81i)17-s + (−2.36 + 4.09i)19-s + (−3.81 − 2.54i)21-s + 0.398·23-s − 2.16·25-s + (3.14 − 4.13i)27-s + (−3.19 + 5.54i)29-s + (−0.289 + 0.500i)31-s + ⋯
L(s)  = 1  + (−0.214 + 0.976i)3-s + 0.752·5-s + (−0.362 + 0.931i)7-s + (−0.907 − 0.419i)9-s − 0.375·11-s + (0.543 + 0.941i)13-s + (−0.161 + 0.735i)15-s + (−0.394 − 0.683i)17-s + (−0.541 + 0.938i)19-s + (−0.832 − 0.554i)21-s + 0.0830·23-s − 0.433·25-s + (0.604 − 0.796i)27-s + (−0.594 + 1.02i)29-s + (−0.0519 + 0.0899i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.924 - 0.381i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093849386\)
\(L(\frac12)\) \(\approx\) \(1.093849386\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.371 - 1.69i)T \)
7 \( 1 + (0.960 - 2.46i)T \)
good5 \( 1 - 1.68T + 5T^{2} \)
11 \( 1 + 1.24T + 11T^{2} \)
13 \( 1 + (-1.96 - 3.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.62 + 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.398T + 23T^{2} \)
29 \( 1 + (3.19 - 5.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.289 - 0.500i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.20 - 7.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.46 - 4.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.212 + 0.368i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.466 + 0.807i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.02 + 5.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.10 + 8.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.70 - 8.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.76 + 4.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.03 + 13.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.03 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.19069823800126169862195514601, −9.407349520944510744474412763583, −9.051308503474322755986598376818, −8.074028781822345281447994692994, −6.61570467976946741947149517313, −5.94370128336553393138931796506, −5.23311244866600420065705885325, −4.20608279500359184610597067059, −3.08675467237573639350427408878, −1.97539709098451610035285227873, 0.48537493991046718078531074115, 1.86158445598562845634187128192, 2.97113320574068348106264537529, 4.28366255952727183542294918849, 5.61064492421870038787778551453, 6.16760713660233436576382216916, 7.04362420586051997437170186269, 7.82607796961513410336020856350, 8.636889639053398954757161791075, 9.667571712434695810882723468216

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