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Properties

Label 2-1008-63.4-c1-0-32
Degree $2$
Conductor $1008$
Sign $0.954 - 0.297i$
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  + (−0.577 + 1.63i)3-s + 1.85·5-s + (2.60 + 0.464i)7-s + (−2.33 − 1.88i)9-s + 2.57·11-s + (2.82 − 4.88i)13-s + (−1.07 + 3.03i)15-s + (3.57 − 6.19i)17-s + (−0.636 − 1.10i)19-s + (−2.26 + 3.98i)21-s − 0.241·23-s − 1.55·25-s + (4.42 − 2.71i)27-s + (0.923 + 1.59i)29-s + (−1.49 − 2.59i)31-s + ⋯
L(s)  = 1  + (−0.333 + 0.942i)3-s + 0.829·5-s + (0.984 + 0.175i)7-s + (−0.777 − 0.628i)9-s + 0.776·11-s + (0.782 − 1.35i)13-s + (−0.276 + 0.782i)15-s + (0.868 − 1.50i)17-s + (−0.146 − 0.252i)19-s + (−0.493 + 0.869i)21-s − 0.0503·23-s − 0.311·25-s + (0.852 − 0.523i)27-s + (0.171 + 0.297i)29-s + (−0.268 − 0.465i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.946711608\)
\(L(\frac12)\) \(\approx\) \(1.946711608\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.577 - 1.63i)T \)
7 \( 1 + (-2.60 - 0.464i)T \)
good5 \( 1 - 1.85T + 5T^{2} \)
11 \( 1 - 2.57T + 11T^{2} \)
13 \( 1 + (-2.82 + 4.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.57 + 6.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.636 + 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.241T + 23T^{2} \)
29 \( 1 + (-0.923 - 1.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.49 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.338 - 0.585i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.733 - 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.14 + 7.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.15 - 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.35 + 5.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.04 - 1.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.47 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 + 4.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.53T + 71T^{2} \)
73 \( 1 + (6.55 - 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.86 - 3.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.00 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.60 - 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.40 - 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.00838075499911766602498017403, −9.302260464298946156954431860948, −8.547081665187649846514934483756, −7.60211276698924681469398625799, −6.31512704244947530517746629431, −5.50932495511104687298056533594, −4.98880380172184933140705161668, −3.79553624321552785153067372523, −2.71664697076430730228942631477, −1.10836755432405381500479511237, 1.54866843671328575616008283099, 1.79563548431039586019042975314, 3.65554237304143949986964902435, 4.79259833255451264763617006889, 5.97230648240622894599207598654, 6.32009433305267421901714728265, 7.36283677356775348802199251322, 8.293672030933643316890923304589, 8.880776740961260700702649609203, 10.01335375001311250077829853166

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