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Properties

Label 2-1008-252.223-c1-0-7
Degree $2$
Conductor $1008$
Sign $-0.492 - 0.870i$
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  + (0.382 − 1.68i)3-s + (−3.07 + 1.77i)5-s + (2.63 − 0.230i)7-s + (−2.70 − 1.29i)9-s + (0.434 + 0.250i)11-s + (−2.36 + 1.36i)13-s + (1.81 + 5.86i)15-s − 1.06i·17-s − 6.16·19-s + (0.618 − 4.54i)21-s + (−3.62 + 2.09i)23-s + (3.78 − 6.56i)25-s + (−3.22 + 4.07i)27-s + (−3.28 + 5.69i)29-s + (1.67 + 2.90i)31-s + ⋯
L(s)  = 1  + (0.220 − 0.975i)3-s + (−1.37 + 0.792i)5-s + (0.996 − 0.0872i)7-s + (−0.902 − 0.430i)9-s + (0.130 + 0.0755i)11-s + (−0.654 + 0.378i)13-s + (0.469 + 1.51i)15-s − 0.258i·17-s − 1.41·19-s + (0.134 − 0.990i)21-s + (−0.755 + 0.436i)23-s + (0.757 − 1.31i)25-s + (−0.619 + 0.784i)27-s + (−0.610 + 1.05i)29-s + (0.300 + 0.521i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.492 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3792008325\)
\(L(\frac12)\) \(\approx\) \(0.3792008325\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 + 1.68i)T \)
7 \( 1 + (-2.63 + 0.230i)T \)
good5 \( 1 + (3.07 - 1.77i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.434 - 0.250i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.36 - 1.36i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.06iT - 17T^{2} \)
19 \( 1 + 6.16T + 19T^{2} \)
23 \( 1 + (3.62 - 2.09i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.28 - 5.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.67 - 2.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.88T + 37T^{2} \)
41 \( 1 + (8.83 - 5.10i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.403 + 0.233i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.56 - 2.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.57T + 53T^{2} \)
59 \( 1 + (-2.30 - 3.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.76 + 1.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.5 - 7.22i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.11iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + (-14.2 - 8.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.96 + 8.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.58iT - 89T^{2} \)
97 \( 1 + (11.1 + 6.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52833003900703752646492945689, −9.170728421334722218937455993923, −8.206785499536928919913689066864, −7.77735591233242028225905674608, −7.04431162382626292555349563665, −6.32572240065365116278860151715, −4.90237378013977312664008845735, −3.94669517254490365886373388962, −2.85398780766500295009901109224, −1.68367398672502600698713500091, 0.16266232635300254706783878409, 2.20884478716871122344259668367, 3.69987620438094125298831490403, 4.40643859461419387673664703958, 4.92486623715363835798255660826, 6.07428592454477548880316315734, 7.62430892287012447201342876587, 8.187976504835266695755819870908, 8.638920502958627381086144759634, 9.650688421390590165681844552358

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