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Properties

Label 2-1008-252.115-c1-0-47
Degree $2$
Conductor $1008$
Sign $-0.281 - 0.959i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + (−1.5 − 0.866i)5-s + (−2 − 1.73i)7-s − 2.99·9-s + (−4.5 + 2.59i)11-s + (4.5 − 2.59i)13-s + (−1.49 + 2.59i)15-s + (1.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−2.99 + 3.46i)21-s + (1.5 + 0.866i)23-s + (−1 − 1.73i)25-s + 5.19i·27-s + (−4.5 + 7.79i)29-s − 8·31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.670 − 0.387i)5-s + (−0.755 − 0.654i)7-s − 0.999·9-s + (−1.35 + 0.783i)11-s + (1.24 − 0.720i)13-s + (−0.387 + 0.670i)15-s + (0.363 + 0.210i)17-s + (0.114 + 0.198i)19-s + (−0.654 + 0.755i)21-s + (0.312 + 0.180i)23-s + (−0.200 − 0.346i)25-s + 0.999i·27-s + (−0.835 + 1.44i)29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.281 - 0.959i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 0.866i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.5 + 4.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.245441064646981944999718019764, −8.287829836525805019695461939140, −7.66811730190439181942107274151, −7.06988547551904641559130919226, −5.99055130334448480568306932760, −5.16596340479139889862269700347, −3.79318405484997809286070410911, −2.93772796253619835626070314014, −1.38416665983028873748409797324, 0, 2.58934417509677849052145739795, 3.43439885485907517267896315833, 4.18268479932770601881380912573, 5.61820790514396981570956004781, 5.91148672145735557669799885502, 7.31076828480359929992406313344, 8.195433997509375173246053614770, 9.035400889505209195548273674686, 9.604658557941998376306732050477

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