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Properties

Label 2-1008-252.115-c1-0-38
Degree $2$
Conductor $1008$
Sign $-0.995 + 0.0986i$
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  + (−1.60 − 0.649i)3-s + (−2.10 − 1.21i)5-s + (0.347 − 2.62i)7-s + (2.15 + 2.08i)9-s + (3.88 − 2.24i)11-s + (0.395 − 0.228i)13-s + (2.59 + 3.32i)15-s + (−1.45 − 0.839i)17-s + (−3.17 − 5.50i)19-s + (−2.26 + 3.98i)21-s + (3.03 + 1.75i)23-s + (0.456 + 0.791i)25-s + (−2.10 − 4.75i)27-s + (−1.38 + 2.39i)29-s + 8.92·31-s + ⋯
L(s)  = 1  + (−0.926 − 0.375i)3-s + (−0.941 − 0.543i)5-s + (0.131 − 0.991i)7-s + (0.718 + 0.695i)9-s + (1.17 − 0.676i)11-s + (0.109 − 0.0633i)13-s + (0.669 + 0.857i)15-s + (−0.352 − 0.203i)17-s + (−0.728 − 1.26i)19-s + (−0.493 + 0.869i)21-s + (0.633 + 0.365i)23-s + (0.0913 + 0.158i)25-s + (−0.405 − 0.914i)27-s + (−0.256 + 0.444i)29-s + 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.995 + 0.0986i$
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: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.995 + 0.0986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6105346257\)
\(L(\frac12)\) \(\approx\) \(0.6105346257\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 0.649i)T \)
7 \( 1 + (-0.347 + 2.62i)T \)
good5 \( 1 + (2.10 + 1.21i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.88 + 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.395 + 0.228i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.45 + 0.839i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.17 + 5.50i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.03 - 1.75i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.38 - 2.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.92T + 31T^{2} \)
37 \( 1 + (-0.463 - 0.802i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.08 - 5.24i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.87 + 5.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.39T + 47T^{2} \)
53 \( 1 + (4.91 - 8.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.45T + 59T^{2} \)
61 \( 1 + 5.31iT - 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 - 1.59iT - 71T^{2} \)
73 \( 1 + (-9.31 - 5.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (0.657 - 1.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-13.5 + 7.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.69 + 3.28i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.608306695372183273384880841656, −8.551210465467380448420033611299, −7.88734876506717673006497482186, −6.76002554309478199261634342078, −6.51226243325637633473156942972, −4.93903165023167109322582228188, −4.46385732539950800129498372320, −3.39939792402340985155246714528, −1.37814307006589853138162315147, −0.34773112412972730609674698614, 1.72085431003394933754437659079, 3.38275789611722780276104659508, 4.23833902704407524416918396507, 5.08725996752858339051660297858, 6.37947582475890327832364155739, 6.61751437471603820928044575767, 7.88572728478237425000250333138, 8.723444121825506926877300815660, 9.688627164762948183233057201404, 10.40285519335397709268806195097

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