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Properties

Label 2-1008-63.25-c1-0-10
Degree $2$
Conductor $1008$
Sign $0.888 - 0.458i$
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  − 1.73i·3-s + (0.5 − 0.866i)5-s + (−2 + 1.73i)7-s − 2.99·9-s + (2.5 + 4.33i)11-s + (2.5 + 4.33i)13-s + (−1.49 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + (2.99 + 3.46i)21-s + (1.5 − 2.59i)23-s + (2 + 3.46i)25-s + 5.19i·27-s + (0.5 − 0.866i)29-s + (7.5 − 4.33i)33-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s − 0.999·9-s + (0.753 + 1.30i)11-s + (0.693 + 1.20i)13-s + (−0.387 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + (0.654 + 0.755i)21-s + (0.312 − 0.541i)23-s + (0.400 + 0.692i)25-s + 0.999i·27-s + (0.0928 − 0.160i)29-s + (1.30 − 0.753i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.375032799\)
\(L(\frac12)\) \(\approx\) \(1.375032799\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.765445825295978126163064632786, −9.072277264127014909531672083455, −8.555833516676425418598792171524, −7.31925582288852351408708995831, −6.59045430726080379694440923603, −6.06806429417226807899383763058, −4.85966648898184933735992438062, −3.69839049767841392856470950817, −2.30278425722012274124389819242, −1.43872806686615462052694172460, 0.66967048072909480403087513491, 3.00247962684438190234538830470, 3.41148916746642944759628897759, 4.50346361805779287416607940372, 5.73610999737650788599225654122, 6.26150769916879585908383218235, 7.35129282530615120844785337826, 8.533563878666205146921791707105, 9.103980445796982240106235260265, 10.00840765282430775524296732333

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