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Properties

Label 2-1008-252.103-c1-0-36
Degree $2$
Conductor $1008$
Sign $0.665 + 0.746i$
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  + (1.62 − 0.608i)3-s + (1.53 − 0.886i)5-s + (−2.64 − 0.0987i)7-s + (2.25 − 1.97i)9-s + (5.33 + 3.08i)11-s + (−4.60 − 2.65i)13-s + (1.95 − 2.37i)15-s + (4.69 − 2.71i)17-s + (0.935 − 1.62i)19-s + (−4.34 + 1.44i)21-s + (−0.562 + 0.324i)23-s + (−0.928 + 1.60i)25-s + (2.46 − 4.57i)27-s + (−1.14 − 1.98i)29-s + 10.2·31-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)3-s + (0.686 − 0.396i)5-s + (−0.999 − 0.0373i)7-s + (0.752 − 0.658i)9-s + (1.60 + 0.929i)11-s + (−1.27 − 0.737i)13-s + (0.503 − 0.612i)15-s + (1.13 − 0.657i)17-s + (0.214 − 0.371i)19-s + (−0.948 + 0.316i)21-s + (−0.117 + 0.0677i)23-s + (−0.185 + 0.321i)25-s + (0.473 − 0.880i)27-s + (−0.213 − 0.369i)29-s + 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.665 + 0.746i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.665 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.411344794\)
\(L(\frac12)\) \(\approx\) \(2.411344794\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 0.608i)T \)
7 \( 1 + (2.64 + 0.0987i)T \)
good5 \( 1 + (-1.53 + 0.886i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.33 - 3.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.60 + 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.69 + 2.71i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.935 + 1.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.562 - 0.324i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.14 + 1.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.64 + 3.83i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.07 - 0.620i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.468T + 47T^{2} \)
53 \( 1 + (0.941 + 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.24T + 59T^{2} \)
61 \( 1 + 6.33iT - 61T^{2} \)
67 \( 1 - 9.93iT - 67T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + (4.77 - 2.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + (-4.06 - 7.03i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.228 + 0.132i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.5 - 7.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.656099771963635876436142274110, −9.367987341278783348400737597742, −8.223006052432968844699841675346, −7.21704871607777351707158041149, −6.72773791311632487047043872221, −5.60310808164514560935437408768, −4.45251811357390654810240800736, −3.34873692844414564595213203265, −2.41316678413393865717219952691, −1.12760026340649337256552601823, 1.59685715384409736663601275906, 2.89138181650514260413252942138, 3.57479024287577463278171436380, 4.63110194049014737262194286554, 6.07748541685837055244674496002, 6.56110290353361558575967094880, 7.61426841227598696374360420606, 8.617751974453242124839482381926, 9.412710633389609402314160560481, 9.878357170287961539952932653144

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