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Properties

Label 2-1008-252.103-c1-0-18
Degree $2$
Conductor $1008$
Sign $0.0595 - 0.998i$
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  + (0.891 + 1.48i)3-s + (2.03 − 1.17i)5-s + (−1.63 + 2.08i)7-s + (−1.40 + 2.64i)9-s + (−0.383 − 0.221i)11-s + (1.96 + 1.13i)13-s + (3.55 + 1.97i)15-s + (4.01 − 2.32i)17-s + (−3.30 + 5.73i)19-s + (−4.54 − 0.569i)21-s + (−0.984 + 0.568i)23-s + (0.257 − 0.445i)25-s + (−5.18 + 0.267i)27-s + (4.37 + 7.58i)29-s + 4.64·31-s + ⋯
L(s)  = 1  + (0.514 + 0.857i)3-s + (0.909 − 0.525i)5-s + (−0.617 + 0.786i)7-s + (−0.469 + 0.882i)9-s + (−0.115 − 0.0667i)11-s + (0.543 + 0.314i)13-s + (0.918 + 0.509i)15-s + (0.974 − 0.562i)17-s + (−0.759 + 1.31i)19-s + (−0.992 − 0.124i)21-s + (−0.205 + 0.118i)23-s + (0.0514 − 0.0891i)25-s + (−0.998 + 0.0515i)27-s + (0.812 + 1.40i)29-s + 0.833·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0595 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0595 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.0595 - 0.998i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.0595 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.003723538\)
\(L(\frac12)\) \(\approx\) \(2.003723538\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.891 - 1.48i)T \)
7 \( 1 + (1.63 - 2.08i)T \)
good5 \( 1 + (-2.03 + 1.17i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.383 + 0.221i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.96 - 1.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.01 + 2.32i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.30 - 5.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.984 - 0.568i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.37 - 7.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.64T + 31T^{2} \)
37 \( 1 + (1.41 - 2.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.55 + 2.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.81 + 2.20i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 + (1.54 + 2.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.37T + 59T^{2} \)
61 \( 1 + 10.2iT - 61T^{2} \)
67 \( 1 + 2.43iT - 67T^{2} \)
71 \( 1 - 7.20iT - 71T^{2} \)
73 \( 1 + (-3.17 + 1.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.38iT - 79T^{2} \)
83 \( 1 + (7.87 + 13.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.32 - 3.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.9 - 8.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.03115216909913554116991137460, −9.351285161922414768328857534114, −8.712022868867057808070060262469, −8.021635809119057895740773748592, −6.57541363379318776463564154540, −5.66152384309727465192534815978, −5.10918945798672451432428825328, −3.82919098244931894288599619829, −2.90180788581626446606095615831, −1.72980520729921995502807488096, 0.888051552803160078792532820361, 2.31178953627040524990941223264, 3.12985878989484082922827742424, 4.28658982555805012178256706401, 5.92143634370025290489889232900, 6.35471084192443753474527800933, 7.17483734375572672963729788704, 8.037856457137273177160768263089, 8.880823247181711801433492584914, 9.907180669081789512992177317792

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