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Properties

Label 2-1008-336.317-c0-0-0
Degree $2$
Conductor $1008$
Sign $0.520 + 0.853i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)10-s + (−0.258 + 0.965i)11-s + (−0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (1.22 − 0.707i)17-s + (1.36 − 0.366i)19-s + (0.707 − 0.707i)20-s + 22-s + (−0.5 + 0.866i)28-s + (0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)10-s + (−0.258 + 0.965i)11-s + (−0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (1.22 − 0.707i)17-s + (1.36 − 0.366i)19-s + (0.707 − 0.707i)20-s + 22-s + (−0.5 + 0.866i)28-s + (0.707 − 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.520 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7946616169\)
\(L(\frac12)\) \(\approx\) \(0.7946616169\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.888449718937173417474251294829, −9.641595206447504613931160066732, −8.149286051571182696003377433018, −7.79440154295634650172692817521, −7.10567145132121772360313072299, −5.29414701656067065890772679838, −4.56601589509895399925881928518, −3.67211504881991137074508897291, −2.64159478866121775166201069601, −1.15263949110931539728575839279, 1.20687041158891062408619798711, 3.30803971102687942955559735781, 4.29152361719826470835145568307, 5.42471332318808695954177381596, 5.79362593803869646315818920745, 7.27298087106655277388033309015, 7.81942251257331022757971680422, 8.460822781283297028231554996465, 9.092704078673100248013850812719, 10.24741599869791183183411320878

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