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Properties

Label 2-1008-9.7-c1-0-9
Degree $2$
Conductor $1008$
Sign $-0.388 - 0.921i$
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  + (1.25 + 1.19i)3-s + (0.846 + 1.46i)5-s + (−0.5 + 0.866i)7-s + (0.150 + 2.99i)9-s + (−0.474 + 0.822i)11-s + (1.69 + 2.93i)13-s + (−0.687 + 2.84i)15-s − 0.300·17-s − 5.73·19-s + (−1.66 + 0.490i)21-s + (−1.17 − 2.03i)23-s + (1.06 − 1.84i)25-s + (−3.38 + 3.93i)27-s + (1.01 − 1.76i)29-s + (0.522 + 0.905i)31-s + ⋯
L(s)  = 1  + (0.724 + 0.689i)3-s + (0.378 + 0.655i)5-s + (−0.188 + 0.327i)7-s + (0.0501 + 0.998i)9-s + (−0.143 + 0.247i)11-s + (0.470 + 0.814i)13-s + (−0.177 + 0.735i)15-s − 0.0729·17-s − 1.31·19-s + (−0.362 + 0.106i)21-s + (−0.244 − 0.423i)23-s + (0.213 − 0.369i)25-s + (−0.651 + 0.758i)27-s + (0.189 − 0.327i)29-s + (0.0938 + 0.162i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.388 - 0.921i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959397918\)
\(L(\frac12)\) \(\approx\) \(1.959397918\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 1.19i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.846 - 1.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.474 - 0.822i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.69 - 2.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.300T + 17T^{2} \)
19 \( 1 + 5.73T + 19T^{2} \)
23 \( 1 + (1.17 + 2.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.01 + 1.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.522 - 0.905i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.22T + 37T^{2} \)
41 \( 1 + (-2.56 - 4.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.25 - 9.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.24 + 5.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.86T + 53T^{2} \)
59 \( 1 + (4.46 + 7.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.256 - 0.443i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.63 - 11.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.04T + 71T^{2} \)
73 \( 1 + 1.59T + 73T^{2} \)
79 \( 1 + (-6.43 + 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.98 + 3.43i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.24T + 89T^{2} \)
97 \( 1 + (5.02 - 8.70i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.13445975354212332639946373896, −9.461549374840658997078586592170, −8.633078155556424420477424353545, −7.953708934311473336439538923280, −6.71948803960194271954077837298, −6.11231149353993408552359311473, −4.76960574025591008630666182312, −4.01106911173486097784761576262, −2.82263934107484695610979056950, −2.05296951908266413716607721219, 0.817942560789594847978793424786, 2.06413228559660619548143299295, 3.23969792929613562036046739164, 4.24323912266529496899976258235, 5.54022685396527428962859325152, 6.32620993439034732530550052624, 7.28917947032760806832484567263, 8.162811929553929043158330373147, 8.755859128369137265266606906789, 9.517110331690216930076889196551

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