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Properties

Label 2-1008-9.7-c1-0-28
Degree $2$
Conductor $1008$
Sign $-0.5 + 0.866i$
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  + (−0.592 − 1.62i)3-s + (0.326 + 0.565i)5-s + (0.5 − 0.866i)7-s + (−2.29 + 1.92i)9-s + (1.70 − 2.95i)11-s + (0.152 + 0.264i)13-s + (0.726 − 0.866i)15-s − 0.226·17-s + 2.16·19-s + (−1.70 − 0.300i)21-s + (−3.35 − 5.80i)23-s + (2.28 − 3.96i)25-s + (4.5 + 2.59i)27-s + (0.254 − 0.441i)29-s + (−2.85 − 4.95i)31-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)3-s + (0.145 + 0.252i)5-s + (0.188 − 0.327i)7-s + (−0.766 + 0.642i)9-s + (0.514 − 0.890i)11-s + (0.0423 + 0.0733i)13-s + (0.187 − 0.223i)15-s − 0.0549·17-s + 0.496·19-s + (−0.372 − 0.0656i)21-s + (−0.698 − 1.21i)23-s + (0.457 − 0.792i)25-s + (0.866 + 0.499i)27-s + (0.0473 − 0.0819i)29-s + (−0.513 − 0.889i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.5 + 0.866i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.231240732\)
\(L(\frac12)\) \(\approx\) \(1.231240732\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.592 + 1.62i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.326 - 0.565i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.70 + 2.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.152 - 0.264i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.226T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 + (3.35 + 5.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.254 + 0.441i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.85 + 4.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.28T + 37T^{2} \)
41 \( 1 + (0.479 + 0.829i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.85 - 6.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.14 + 7.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.18T + 53T^{2} \)
59 \( 1 + (6.07 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.75 + 3.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.87 - 4.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + (5.98 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.13 + 3.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.19T + 89T^{2} \)
97 \( 1 + (-7.48 + 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.748501614592338143786269350186, −8.588790896747539824538745797729, −8.059755728187634818648380944324, −7.01791702367501891681373416126, −6.37152504317687737908355031266, −5.61131980679155990771931140700, −4.42781065817627612562104314439, −3.15342914830882326567102860683, −1.96972346290567967426174277039, −0.60438639838652310942737813186, 1.58106444922719326418565708063, 3.14979542044860197582821434827, 4.12906842990916666104692781190, 5.07941971579045340154280058969, 5.69498971446492775098182229093, 6.80203034293199513161915046874, 7.76983878583439068020567706186, 8.982653614048956156415974895957, 9.322894028891219339682114036858, 10.20357561601230489608249034414

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