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Properties

Label 2-1008-252.31-c1-0-26
Degree $2$
Conductor $1008$
Sign $0.996 + 0.0862i$
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 + 0.717i·5-s + (1.62 + 2.09i)7-s + (1.5 − 2.59i)9-s − 1.73i·11-s + (3.62 + 2.09i)13-s + (0.621 + 1.07i)15-s + (−2.74 − 1.58i)17-s + (0.5 + 0.866i)19-s + (4.24 + 1.73i)21-s + 2.74i·23-s + 4.48·25-s − 5.19i·27-s + (0.621 + 1.07i)29-s + (−2 − 3.46i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + 0.320i·5-s + (0.612 + 0.790i)7-s + (0.5 − 0.866i)9-s − 0.522i·11-s + (1.00 + 0.579i)13-s + (0.160 + 0.277i)15-s + (−0.665 − 0.384i)17-s + (0.114 + 0.198i)19-s + (0.925 + 0.377i)21-s + 0.572i·23-s + 0.897·25-s − 0.999i·27-s + (0.115 + 0.199i)29-s + (−0.359 − 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.996 + 0.0862i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.996 + 0.0862i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.415805328\)
\(L(\frac12)\) \(\approx\) \(2.415805328\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
good5 \( 1 - 0.717iT - 5T^{2} \)
11 \( 1 + 1.73iT - 11T^{2} \)
13 \( 1 + (-3.62 - 2.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.74 + 1.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.74iT - 23T^{2} \)
29 \( 1 + (-0.621 - 1.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.74 - 5.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.74 - 3.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.24 + 7.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.621 - 1.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.75 - 1.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.74 + 9.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (14.2 - 8.21i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.74 + 3.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.686639087677233519138041981441, −8.922435036878865869291952960451, −8.423039171619323585334585156851, −7.55811230050890886774511042675, −6.60763339465528680760884743295, −5.85567880251763762219836403259, −4.57234564297452348214584034609, −3.44849115990989611222559157105, −2.50050965034555916865752689139, −1.39194112199450662851100491356, 1.28909731602658871808620712879, 2.60564634505660515025632565159, 3.87315565694976427759297664122, 4.46520523520664842589658749324, 5.45048618529686864389697415919, 6.80974553855314858820398758567, 7.62298323539855649434383871201, 8.479716613649731953062449461639, 8.946379025948332111986667573245, 10.02845029878535939604319909301

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