↑

Properties

Label 2-1008-252.11-c1-0-1
Degree $2$
Conductor $1008$
Sign $-0.990 + 0.135i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.743 + 1.56i)3-s + (−0.126 − 0.0730i)5-s + (−0.562 − 2.58i)7-s + (−1.89 − 2.32i)9-s + (2.18 + 3.78i)11-s + (0.804 + 1.39i)13-s + (0.208 − 0.143i)15-s + (−4.71 − 2.72i)17-s + (−4.28 + 2.47i)19-s + (4.46 + 1.04i)21-s + (−4.26 + 7.39i)23-s + (−2.48 − 4.31i)25-s + (5.04 − 1.23i)27-s + (−0.394 − 0.227i)29-s + 6.13i·31-s + ⋯
L(s)  = 1  + (−0.429 + 0.903i)3-s + (−0.0566 − 0.0326i)5-s + (−0.212 − 0.977i)7-s + (−0.631 − 0.775i)9-s + (0.659 + 1.14i)11-s + (0.223 + 0.386i)13-s + (0.0538 − 0.0370i)15-s + (−1.14 − 0.660i)17-s + (−0.983 + 0.567i)19-s + (0.973 + 0.227i)21-s + (−0.890 + 1.54i)23-s + (−0.497 − 0.862i)25-s + (0.971 − 0.237i)27-s + (−0.0732 − 0.0423i)29-s + 1.10i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.990 + 0.135i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3010221223\)
\(L(\frac12)\) \(\approx\) \(0.3010221223\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.743 - 1.56i)T \)
7 \( 1 + (0.562 + 2.58i)T \)
good5 \( 1 + (0.126 + 0.0730i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.804 - 1.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.71 + 2.72i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.28 - 2.47i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.26 - 7.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.394 + 0.227i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.13iT - 31T^{2} \)
37 \( 1 + (0.739 + 1.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.15 + 0.664i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.33 + 4.23i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 + (-3.76 - 2.17i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.02T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 9.33iT - 67T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 + (-3.45 + 5.99i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.640iT - 79T^{2} \)
83 \( 1 + (-1.54 + 2.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.58 - 1.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.24 - 9.08i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.33276087209277154722402715455, −9.665419679072587981856546723889, −9.030171771580478770866453297346, −7.895869717592196205342481504889, −6.82162915149164344935789586158, −6.28453143694693184537160407230, −4.93792118971782089168473658842, −4.23108580034550186717148152097, −3.57377146187979030244240544463, −1.82453370696510557652478281205, 0.13733865966899852122397841692, 1.84482009022321752580778966185, 2.84916057018315255877904306766, 4.21525163249671885698111814045, 5.48003104100928464653758049666, 6.33684900749475961398262726826, 6.57938480150186788149850710649, 8.138744496743348015430997066532, 8.453211906748839581880557817319, 9.338248813826945903683659411868

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