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Properties

Label 2-1008-63.47-c1-0-44
Degree $2$
Conductor $1008$
Sign $-0.992 - 0.125i$
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  + (−0.127 − 1.72i)3-s + 2.18·5-s + (−2.64 + 0.0736i)7-s + (−2.96 + 0.441i)9-s + 1.46i·11-s + (−2.92 + 1.69i)13-s + (−0.278 − 3.77i)15-s + (−1.32 − 2.28i)17-s + (−6.87 − 3.97i)19-s + (0.465 + 4.55i)21-s − 4.00i·23-s − 0.234·25-s + (1.14 + 5.06i)27-s + (−6.71 − 3.87i)29-s + (−0.612 − 0.353i)31-s + ⋯
L(s)  = 1  + (−0.0737 − 0.997i)3-s + 0.976·5-s + (−0.999 + 0.0278i)7-s + (−0.989 + 0.147i)9-s + 0.441i·11-s + (−0.811 + 0.468i)13-s + (−0.0720 − 0.973i)15-s + (−0.320 − 0.555i)17-s + (−1.57 − 0.911i)19-s + (0.101 + 0.994i)21-s − 0.836i·23-s − 0.0468·25-s + (0.219 + 0.975i)27-s + (−1.24 − 0.719i)29-s + (−0.109 − 0.0634i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5058326849\)
\(L(\frac12)\) \(\approx\) \(0.5058326849\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.127 + 1.72i)T \)
7 \( 1 + (2.64 - 0.0736i)T \)
good5 \( 1 - 2.18T + 5T^{2} \)
11 \( 1 - 1.46iT - 11T^{2} \)
13 \( 1 + (2.92 - 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.32 + 2.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.87 + 3.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + (6.71 + 3.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.612 + 0.353i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.41 + 2.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.74 + 6.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.27 + 2.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.27 - 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.41 + 1.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.71 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.75 + 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.92 + 5.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (3.95 - 2.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.69 - 8.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.70 + 2.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.61 - 8.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.38 + 3.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.321074037238775901514472006100, −8.974797688788164960308891984165, −7.63838171068781919707095940811, −6.84137310573743938933019556832, −6.31323822324405519839292035311, −5.44784666798620371604469714387, −4.23257291178239750594334795803, −2.56353855784794482983301017507, −2.11502141651825166658853621136, −0.20577873416860980763093620521, 2.10657810172360182283592073173, 3.26900031868214533871216670605, 4.12413564973084561680669848825, 5.42296009469180792608460919751, 5.91165332420252227475803317791, 6.78276603722679792498068690607, 8.133173986794906349863686747969, 8.993296048514129297860141085639, 9.722315676718159005041196744290, 10.22130429171424899938598068752

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