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Properties

Label 2-1008-12.11-c1-0-3
Degree $2$
Conductor $1008$
Sign $-0.0917 - 0.995i$
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  + 3.86i·5-s + i·7-s + 5.27·11-s + 2·13-s + 1.03i·17-s − 5.46i·19-s + 0.378·23-s − 9.92·25-s + 6.31i·29-s + 9.46i·31-s − 3.86·35-s − 10.9·37-s + 5.93i·41-s + 4i·43-s + 8.48·47-s + ⋯
L(s)  = 1  + 1.72i·5-s + 0.377i·7-s + 1.59·11-s + 0.554·13-s + 0.251i·17-s − 1.25i·19-s + 0.0790·23-s − 1.98·25-s + 1.17i·29-s + 1.69i·31-s − 0.653·35-s − 1.79·37-s + 0.926i·41-s + 0.609i·43-s + 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0917 - 0.995i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.0917 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663628001\)
\(L(\frac12)\) \(\approx\) \(1.663628001\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 3.86iT - 5T^{2} \)
11 \( 1 - 5.27T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.03iT - 17T^{2} \)
19 \( 1 + 5.46iT - 19T^{2} \)
23 \( 1 - 0.378T + 23T^{2} \)
29 \( 1 - 6.31iT - 29T^{2} \)
31 \( 1 - 9.46iT - 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 5.93iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 3.46T + 61T^{2} \)
67 \( 1 + 15.4iT - 67T^{2} \)
71 \( 1 + 4.52T + 71T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 3.86iT - 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−10.42717547822207812572303553388, −9.248635650029616562007303553545, −8.733023255219686584228154074792, −7.42333672324379781142074520351, −6.63558125176530635589335008936, −6.35007512931430215602049707083, −4.97695272468175226707652098288, −3.62682589717484680371591438590, −3.04649476514689913834087011973, −1.67878221185208099420586179220, 0.830374231802114226653387843644, 1.80691232043269650224597286013, 3.89930167377660928251689358373, 4.18411049522223509031620944185, 5.46913007787144341517346949418, 6.14241584938693963088013465789, 7.34280191211232291904499119700, 8.268617038622343057855589274916, 8.955373274219018990707985710724, 9.510070202362931138620399488757

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