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Properties

Label 2-1008-48.35-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.981 - 0.189i$
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  + (−1.16 − 0.795i)2-s + (0.735 + 1.85i)4-s + (1.41 + 1.41i)5-s − 7-s + (0.619 − 2.75i)8-s + (−0.529 − 2.77i)10-s + (0.748 − 0.748i)11-s + (4.24 + 4.24i)13-s + (1.16 + 0.795i)14-s + (−2.91 + 2.73i)16-s − 7.50i·17-s + (−1.59 + 3.67i)20-s + (−1.47 + 0.280i)22-s + 4.59i·23-s − 0.999i·25-s + (−1.59 − 8.34i)26-s + ⋯
L(s)  = 1  + (−0.826 − 0.562i)2-s + (0.367 + 0.929i)4-s + (0.632 + 0.632i)5-s − 0.377·7-s + (0.218 − 0.975i)8-s + (−0.167 − 0.878i)10-s + (0.225 − 0.225i)11-s + (1.17 + 1.17i)13-s + (0.312 + 0.212i)14-s + (−0.729 + 0.683i)16-s − 1.82i·17-s + (−0.355 + 0.820i)20-s + (−0.313 + 0.0597i)22-s + 0.958i·23-s − 0.199i·25-s + (−0.311 − 1.63i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218311307\)
\(L(\frac12)\) \(\approx\) \(1.218311307\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.795i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \)
11 \( 1 + (-0.748 + 0.748i)T - 11iT^{2} \)
13 \( 1 + (-4.24 - 4.24i)T + 13iT^{2} \)
17 \( 1 + 7.50iT - 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 - 4.59iT - 23T^{2} \)
29 \( 1 + (-5.26 + 5.26i)T - 29iT^{2} \)
31 \( 1 - 8.49iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 - 0.978T + 41T^{2} \)
43 \( 1 + (-6.80 - 6.80i)T + 43iT^{2} \)
47 \( 1 + 6.36T + 47T^{2} \)
53 \( 1 + (-8.17 - 8.17i)T + 53iT^{2} \)
59 \( 1 + (4.51 - 4.51i)T - 59iT^{2} \)
61 \( 1 + (-0.249 - 0.249i)T + 61iT^{2} \)
67 \( 1 + (-0.750 + 0.750i)T - 67iT^{2} \)
71 \( 1 - 9.62iT - 71T^{2} \)
73 \( 1 - 8.61iT - 73T^{2} \)
79 \( 1 + 16.9iT - 79T^{2} \)
83 \( 1 + (-2.47 - 2.47i)T + 83iT^{2} \)
89 \( 1 + 2.55T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−9.928640581895284677016788917185, −9.253132225489389187041587246048, −8.662191532618392191967218501525, −7.46213004153637536888710054087, −6.71861055816926140668798320764, −6.05466687240540710158851305484, −4.49962584744195208239102484563, −3.32975901921270759593246185473, −2.50224059696230831890203768439, −1.18528802687487021233464075911, 0.896557567337346240915534447848, 2.05880581506517745330031173653, 3.65577338935906540885050444092, 5.01405740627429788331967247243, 5.99367219653406317704172368642, 6.33383475118225097192779150122, 7.60044985953372485442492522858, 8.516823802680569226772384579999, 8.849205644216716918280477659486, 9.940639342096271416347091510124

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