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Properties

Label 2-1008-63.16-c1-0-34
Degree $2$
Conductor $1008$
Sign $0.678 + 0.734i$
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  + (1.5 + 0.866i)3-s − 3·5-s + (0.5 − 2.59i)7-s + (1.5 + 2.59i)9-s + 3·11-s + (−2.5 − 4.33i)13-s + (−4.5 − 2.59i)15-s + (−1.5 − 2.59i)17-s + (2.5 − 4.33i)19-s + (3 − 3.46i)21-s + 3·23-s + 4·25-s + 5.19i·27-s + (1.5 − 2.59i)29-s + (−2 + 3.46i)31-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s − 1.34·5-s + (0.188 − 0.981i)7-s + (0.5 + 0.866i)9-s + 0.904·11-s + (−0.693 − 1.20i)13-s + (−1.16 − 0.670i)15-s + (−0.363 − 0.630i)17-s + (0.573 − 0.993i)19-s + (0.654 − 0.755i)21-s + 0.625·23-s + 0.800·25-s + 0.999i·27-s + (0.278 − 0.482i)29-s + (−0.359 + 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640489022\)
\(L(\frac12)\) \(\approx\) \(1.640489022\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.734345875876917233596344431059, −9.042822951602141620102967007767, −8.074092896059885839960909244989, −7.47304505439843099005684172270, −6.93748198389936398608138190789, −5.13478383096915596308704737904, −4.35205902404046042657216957310, −3.63062489716468174536171999250, −2.71786789828388656733516991111, −0.73773056434073835186735520205, 1.48935810052914854352440585123, 2.70386956602701105077126426919, 3.82498991891768496787275847440, 4.47062198330237167266321499924, 5.97904267187937986172165221942, 6.94638928566127237028614098593, 7.62815094030494200282008483683, 8.440684370648498443989306083893, 9.042344611374360403096746060272, 9.742829372810773796081022229767

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