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Properties

Label 2-1008-28.11-c2-0-25
Degree $2$
Conductor $1008$
Sign $0.978 - 0.204i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.53 + 4.38i)5-s + (6.77 + 1.76i)7-s + (−13.3 − 7.70i)11-s + 8.82·13-s + (10.8 − 18.7i)17-s + (8.68 − 5.01i)19-s + (20.9 − 12.0i)23-s + (−0.312 + 0.541i)25-s + 36.8·29-s + (−3.14 − 1.81i)31-s + (9.42 + 34.1i)35-s + (29.5 + 51.1i)37-s − 63.5·41-s − 30.9i·43-s + (17.2 − 9.97i)47-s + ⋯
L(s)  = 1  + (0.506 + 0.876i)5-s + (0.967 + 0.251i)7-s + (−1.21 − 0.700i)11-s + 0.678·13-s + (0.636 − 1.10i)17-s + (0.457 − 0.263i)19-s + (0.910 − 0.525i)23-s + (−0.0125 + 0.0216i)25-s + 1.26·29-s + (−0.101 − 0.0586i)31-s + (0.269 + 0.975i)35-s + (0.797 + 1.38i)37-s − 1.54·41-s − 0.719i·43-s + (0.367 − 0.212i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.978 - 0.204i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.978 - 0.204i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.431216092\)
\(L(\frac12)\) \(\approx\) \(2.431216092\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (-6.77 - 1.76i)T \)
good5 \( 1 + (-2.53 - 4.38i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (13.3 + 7.70i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 8.82T + 169T^{2} \)
17 \( 1 + (-10.8 + 18.7i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.68 + 5.01i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-20.9 + 12.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 36.8T + 841T^{2} \)
31 \( 1 + (3.14 + 1.81i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-29.5 - 51.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 63.5T + 1.68e3T^{2} \)
43 \( 1 + 30.9iT - 1.84e3T^{2} \)
47 \( 1 + (-17.2 + 9.97i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (24.1 - 41.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (79.8 + 46.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-53.3 - 92.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-31.2 - 18.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + (8.21 - 14.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-60.6 + 35.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 13.0iT - 6.88e3T^{2} \)
89 \( 1 + (-5.06 - 8.76i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 117.T + 9.40e3T^{2} \)
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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

−10.04126963538746556860837067478, −8.827723454595560348851061786592, −8.177986917305454319113519375503, −7.29387837132594531259230709106, −6.39041048779486954656295410379, −5.42097550883458835600932033472, −4.76810093622389627979902734299, −3.12302951738490347715655610026, −2.55774542325032739871207711524, −0.996470681493763899424064785331, 1.07600618851363545663806377944, 1.97069624632319629542390870288, 3.46033258828032527545162036291, 4.76834864677012361621381942328, 5.20023337376849427163110348353, 6.17166532536794318814230874389, 7.48633190225518136828804034935, 8.077514739263096421590665954247, 8.830887373047510719884039452574, 9.778736489060870573944511573011

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