Properties

Label 2-1008-252.103-c1-0-44
Degree $2$
Conductor $1008$
Sign $-0.690 - 0.723i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.690 - 0.723i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.690 - 0.723i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−9.771723215978993452367705885833, −8.334228411483683157152767648286, −7.64485300205579036115538510653, −6.95427890014940484405631502180, −5.98103318494248854148557399498, −5.22125234839579588255596246994, −4.21539192318077838226345494506, −2.94806202445393416291259000576, −1.40780492243388931553261632538, 0, 2.07987778364118918133099247406, 3.30297762789591676320546996660, 4.67458007972770686813472574152, 5.19968429157954576102283825472, 6.14341443072050005131946147359, 6.94572249183600658345164168460, 8.001353158581588686890231171607, 9.088472285284999150630395869925, 9.656113499038128208853579956664

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