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Properties

Label 2-1008-252.115-c1-0-40
Degree $2$
Conductor $1008$
Sign $0.281 + 0.959i$
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 + (0.5 − 2.59i)7-s + (1.5 − 2.59i)9-s + (3 − 1.73i)11-s + (−1.5 + 0.866i)13-s + (1.5 + 0.866i)17-s + (2.5 + 4.33i)19-s + (−1.5 − 4.33i)21-s + (−3 − 1.73i)23-s + (−2.5 − 4.33i)25-s − 5.19i·27-s + (−1.5 + 2.59i)29-s − 31-s + (3 − 5.19i)33-s + (−3.5 − 6.06i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.188 − 0.981i)7-s + (0.5 − 0.866i)9-s + (0.904 − 0.522i)11-s + (−0.416 + 0.240i)13-s + (0.363 + 0.210i)17-s + (0.573 + 0.993i)19-s + (−0.327 − 0.944i)21-s + (−0.625 − 0.361i)23-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (−0.278 + 0.482i)29-s − 0.179·31-s + (0.522 − 0.904i)33-s + (−0.575 − 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.244068717\)
\(L(\frac12)\) \(\approx\) \(2.244068717\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 15T + 59T^{2} \)
61 \( 1 + 1.73iT - 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.837147548740087360524857288925, −8.821488985283046628773388187527, −8.132590810976590708295117695620, −7.32063871918843431211814253998, −6.65791770779633399738343930165, −5.58985726301466298688621424048, −4.03601918940879559176678853399, −3.66367562909593689065816477613, −2.17742987794193203769958104679, −1.00566263904630957776191389759, 1.76862279691920699171063195345, 2.79519533687752134454813965518, 3.81101404945237639188635065524, 4.86595464522402419086456869466, 5.65013247888171610059278431792, 6.95969319067565421437770103243, 7.73371322268840453574308951633, 8.623244163171457247585221814413, 9.425171244275730096917465766313, 9.724511980708443519815528802470

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