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Properties

Label 2-1008-252.103-c1-0-28
Degree $2$
Conductor $1008$
Sign $0.642 + 0.766i$
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  + (−1.72 − 0.128i)3-s + (2.43 − 1.40i)5-s + (−0.717 − 2.54i)7-s + (2.96 + 0.444i)9-s + (2.29 + 1.32i)11-s + (4.59 + 2.65i)13-s + (−4.38 + 2.11i)15-s + (−2.35 + 1.36i)17-s + (0.274 − 0.474i)19-s + (0.911 + 4.49i)21-s + (1.98 − 1.14i)23-s + (1.45 − 2.51i)25-s + (−5.06 − 1.15i)27-s + (−3.72 − 6.45i)29-s + 2.76·31-s + ⋯
L(s)  = 1  + (−0.997 − 0.0743i)3-s + (1.08 − 0.628i)5-s + (−0.271 − 0.962i)7-s + (0.988 + 0.148i)9-s + (0.690 + 0.398i)11-s + (1.27 + 0.735i)13-s + (−1.13 + 0.546i)15-s + (−0.571 + 0.329i)17-s + (0.0628 − 0.108i)19-s + (0.198 + 0.980i)21-s + (0.413 − 0.238i)23-s + (0.290 − 0.503i)25-s + (−0.975 − 0.221i)27-s + (−0.692 − 1.19i)29-s + 0.497·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.642 + 0.766i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.495429219\)
\(L(\frac12)\) \(\approx\) \(1.495429219\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.128i)T \)
7 \( 1 + (0.717 + 2.54i)T \)
good5 \( 1 + (-2.43 + 1.40i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.29 - 1.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.59 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.35 - 1.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.274 + 0.474i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.98 + 1.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.72 + 6.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.76T + 31T^{2} \)
37 \( 1 + (-1.81 + 3.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.12 - 4.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.93 - 2.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + (2.53 + 4.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (-4.29 + 2.47i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.61iT - 79T^{2} \)
83 \( 1 + (0.719 + 1.24i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.24 - 1.87i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.7 - 9.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.624921936124591921525230110851, −9.468700572892167631167850605428, −8.205225912175293744302644936911, −6.97451362754615617989300754757, −6.38261959925066765125893348623, −5.71955645151651253500259427771, −4.53829867665234404530704209447, −3.93907881303778106107756117699, −1.89794560138129249781254047656, −0.948205267961119837591635565972, 1.27458453920667622395337606988, 2.63589485026228676838819875784, 3.80949522542363355737583460475, 5.24090238993116571010624017272, 5.93025337647902263749070452228, 6.33470307745286715695354700624, 7.26734888936066997474764880374, 8.733498726695902107846331205773, 9.271286335420661495228166294829, 10.24398028513506077827354859679

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