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Properties

Label 2-1008-63.59-c1-0-26
Degree $2$
Conductor $1008$
Sign $0.861 - 0.507i$
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  + (−0.626 + 1.61i)3-s + 3.55·5-s + (1.49 − 2.18i)7-s + (−2.21 − 2.02i)9-s + 3.02i·11-s + (0.888 + 0.513i)13-s + (−2.22 + 5.73i)15-s + (0.809 − 1.40i)17-s + (7.12 − 4.11i)19-s + (2.58 + 3.78i)21-s − 3.35i·23-s + 7.61·25-s + (4.65 − 2.30i)27-s + (−3.70 + 2.13i)29-s + (−5.18 + 2.99i)31-s + ⋯
L(s)  = 1  + (−0.361 + 0.932i)3-s + 1.58·5-s + (0.564 − 0.825i)7-s + (−0.738 − 0.674i)9-s + 0.911i·11-s + (0.246 + 0.142i)13-s + (−0.574 + 1.48i)15-s + (0.196 − 0.339i)17-s + (1.63 − 0.943i)19-s + (0.565 + 0.824i)21-s − 0.700i·23-s + 1.52·25-s + (0.895 − 0.444i)27-s + (−0.687 + 0.397i)29-s + (−0.931 + 0.537i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.861 - 0.507i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.036622762\)
\(L(\frac12)\) \(\approx\) \(2.036622762\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.626 - 1.61i)T \)
7 \( 1 + (-1.49 + 2.18i)T \)
good5 \( 1 - 3.55T + 5T^{2} \)
11 \( 1 - 3.02iT - 11T^{2} \)
13 \( 1 + (-0.888 - 0.513i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.809 + 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.12 + 4.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.35iT - 23T^{2} \)
29 \( 1 + (3.70 - 2.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.18 - 2.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.92 - 5.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0472 - 0.0817i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.05 + 5.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.57 + 4.45i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.76 - 1.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.42 - 7.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.06 + 2.34i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.187 + 0.325i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-1.13 - 0.655i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.462 + 0.800i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.43 - 9.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.35 - 4.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.3 - 7.69i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.970531546865623461832351475885, −9.485775627572941408939272213876, −8.718380331813144041113136169569, −7.31380088536268794857238753859, −6.61764437472785222142549869416, −5.35632388466856561355816537011, −5.08926485825118211695136217192, −3.93011918137925250048204219136, −2.63034892983714063230346651007, −1.25976103730187911795230654962, 1.31041207747980465225106562648, 2.10044019997712020117493553285, 3.25727086151644181599991334074, 5.20674788385239550242618139238, 5.82758001471901384647835643448, 6.04693906362197400018403753152, 7.43643440257349938449637387201, 8.151216642740362321773645640024, 9.132408062495751576379362171697, 9.750625620028818004796342152567

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