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Properties

Label 2-1008-28.3-c1-0-0
Degree $2$
Conductor $1008$
Sign $-0.978 - 0.205i$
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  + (−1.5 + 0.866i)5-s + (0.5 + 2.59i)7-s + (−1.5 − 0.866i)11-s + (−3 − 1.73i)17-s + (1 + 1.73i)19-s + (−1 + 1.73i)25-s − 9·29-s + (−2.5 + 4.33i)31-s + (−3 − 3.46i)35-s + (−5 − 8.66i)37-s − 10.3i·41-s − 3.46i·43-s + (6 + 10.3i)47-s + (−6.5 + 2.59i)49-s + (−4.5 + 7.79i)53-s + ⋯
L(s)  = 1  + (−0.670 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.452 − 0.261i)11-s + (−0.727 − 0.420i)17-s + (0.229 + 0.397i)19-s + (−0.200 + 0.346i)25-s − 1.67·29-s + (−0.449 + 0.777i)31-s + (−0.507 − 0.585i)35-s + (−0.821 − 1.42i)37-s − 1.62i·41-s − 0.528i·43-s + (0.875 + 1.51i)47-s + (−0.928 + 0.371i)49-s + (−0.618 + 1.07i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.978 - 0.205i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.978 - 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4387126835\)
\(L(\frac12)\) \(\approx\) \(0.4387126835\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (3 + 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12 + 6.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (-3 + 1.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−10.63003244423538236457362486274, −9.196584443765097739552033397841, −8.912671000369812131225293491379, −7.67899035872686296459908945376, −7.25115737345175603387913699904, −5.92891124247341074304070858059, −5.31356309469058832063843697828, −4.07643661437941303177221755672, −3.08870515957561331948077713993, −1.97355284828027452590557258020, 0.19068555869781792360466605670, 1.79801334137394120496675071308, 3.35970825456074349191837411546, 4.29048662291944061885694971610, 4.97978908789291526217131863315, 6.24437818555985477889632961342, 7.23406371073138902944668589510, 7.85402883432371423855138908813, 8.623437817106801739532009345320, 9.648379453631351592936606508207

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