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Properties

Label 2-1008-63.4-c1-0-25
Degree $2$
Conductor $1008$
Sign $0.816 - 0.577i$
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  + (1.58 + 0.691i)3-s + 1.33·5-s + (2.54 − 0.728i)7-s + (2.04 + 2.19i)9-s − 1.51·11-s + (−2.58 + 4.48i)13-s + (2.11 + 0.923i)15-s + (0.774 − 1.34i)17-s + (1.25 + 2.16i)19-s + (4.54 + 0.602i)21-s + 7.36·23-s − 3.21·25-s + (1.72 + 4.90i)27-s + (−0.0309 − 0.0536i)29-s + (−1.92 − 3.33i)31-s + ⋯
L(s)  = 1  + (0.916 + 0.399i)3-s + 0.596·5-s + (0.961 − 0.275i)7-s + (0.681 + 0.732i)9-s − 0.456·11-s + (−0.717 + 1.24i)13-s + (0.547 + 0.238i)15-s + (0.187 − 0.325i)17-s + (0.287 + 0.497i)19-s + (0.991 + 0.131i)21-s + 1.53·23-s − 0.643·25-s + (0.332 + 0.943i)27-s + (−0.00575 − 0.00996i)29-s + (−0.345 − 0.598i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.816 - 0.577i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.633355855\)
\(L(\frac12)\) \(\approx\) \(2.633355855\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 0.691i)T \)
7 \( 1 + (-2.54 + 0.728i)T \)
good5 \( 1 - 1.33T + 5T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.25 - 2.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.36T + 23T^{2} \)
29 \( 1 + (0.0309 + 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.92 + 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.51 + 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.75 - 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.755 + 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.22 + 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.95 - 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.80 + 4.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.703 - 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.832729664484681562264082920350, −9.310104229846700684166082908922, −8.468169181228294631232734842346, −7.57100676891821570087818366846, −6.97256099725370534230195272627, −5.47423076962454437391833242612, −4.76504180526956322444457831673, −3.82448848951407159551220755668, −2.51265368861171453944512369271, −1.66500929920825044480752818406, 1.27269440005808416722005391436, 2.44543687407955493195488294677, 3.23210338918224989513835238754, 4.76150318421639069198478826748, 5.44229624734176223670652972172, 6.62579355612591342004188738427, 7.66048096771391285124215819182, 8.083272204467138337447701558360, 9.019900112725486084034966890532, 9.759969787367577163870344112266

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