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Properties

Label 2-1008-336.221-c0-0-0
Degree $2$
Conductor $1008$
Sign $-0.235 - 0.971i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)10-s + (−0.965 + 0.258i)11-s + (0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (1.22 + 0.707i)17-s + (−0.366 + 1.36i)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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)10-s + (−0.965 + 0.258i)11-s + (0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (1.22 + 0.707i)17-s + (−0.366 + 1.36i)19-s + (−0.707 + 0.707i)20-s + 22-s + (−0.5 − 0.866i)28-s + (−0.707 + 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ -0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3951405952\)
\(L(\frac12)\) \(\approx\) \(0.3951405952\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
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

−10.33762121247339810444323607812, −9.937033152511023276909316741319, −8.727005646024184924216324609122, −7.86494969014710909129477761660, −7.20386371605226071639033124547, −6.51651851318415494944915735611, −5.47258109370917244669335647538, −3.55893614950757876022507068500, −3.26167031981951380052890179966, −1.78564265226672449530260624586, 0.48373112927566224127392078059, 2.27132686108002357233558864137, 3.36479585519464963054673514120, 5.07665417946787275491451218101, 5.58667483158873580122619502611, 6.75079383930462454828710365249, 7.53923537721704589526147367135, 8.491810230572996538874992475812, 8.953796933339175293046356579419, 9.823475460976901130910761640827

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