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Properties

Label 2-1008-252.103-c1-0-27
Degree $2$
Conductor $1008$
Sign $0.732 - 0.680i$
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  + (1.42 + 0.987i)3-s + (−0.679 + 0.392i)5-s + (2.49 − 0.891i)7-s + (1.04 + 2.81i)9-s + (1.22 + 0.708i)11-s + (−1.50 − 0.868i)13-s + (−1.35 − 0.112i)15-s + (5.43 − 3.13i)17-s + (−0.736 + 1.27i)19-s + (4.42 + 1.19i)21-s + (4.85 − 2.80i)23-s + (−2.19 + 3.79i)25-s + (−1.28 + 5.03i)27-s + (3.95 + 6.85i)29-s − 8.41·31-s + ⋯
L(s)  = 1  + (0.821 + 0.570i)3-s + (−0.303 + 0.175i)5-s + (0.941 − 0.337i)7-s + (0.349 + 0.936i)9-s + (0.369 + 0.213i)11-s + (−0.417 − 0.240i)13-s + (−0.349 − 0.0291i)15-s + (1.31 − 0.761i)17-s + (−0.168 + 0.292i)19-s + (0.965 + 0.259i)21-s + (1.01 − 0.584i)23-s + (−0.438 + 0.759i)25-s + (−0.246 + 0.969i)27-s + (0.734 + 1.27i)29-s − 1.51·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.732 - 0.680i$
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: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.318021664\)
\(L(\frac12)\) \(\approx\) \(2.318021664\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 0.987i)T \)
7 \( 1 + (-2.49 + 0.891i)T \)
good5 \( 1 + (0.679 - 0.392i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.22 - 0.708i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.50 + 0.868i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.43 + 3.13i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.736 - 1.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.85 + 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.95 - 6.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.41T + 31T^{2} \)
37 \( 1 + (-3.74 + 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.19 - 4.15i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.85 - 4.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.110T + 47T^{2} \)
53 \( 1 + (4.28 + 7.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.0736T + 59T^{2} \)
61 \( 1 - 1.23iT - 61T^{2} \)
67 \( 1 - 11.8iT - 67T^{2} \)
71 \( 1 + 0.390iT - 71T^{2} \)
73 \( 1 + (-3.70 + 2.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.00iT - 79T^{2} \)
83 \( 1 + (7.88 + 13.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.15 - 3.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.89 + 3.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.968031104052138705406900292681, −9.263666190543343215497944925661, −8.392582114350012130851398549157, −7.58646355180863394774696786654, −7.10419033973690389380360696495, −5.43943820741144038215369744887, −4.75511454840931610669342368419, −3.75324984798919823661461357347, −2.85505613048754468228268832233, −1.46707695576614087151239669439, 1.18010464047348220113880488457, 2.28958113299041923527232257941, 3.48533098064009237454755198774, 4.45191393642857653314023841143, 5.59260258176146027662250905787, 6.59706641771964512213071210535, 7.70076412419560977574833757601, 8.034913374039275743814177202227, 8.935771213761137969334358204779, 9.622253638991882780709154049005

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