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Properties

Label 2-1008-252.103-c1-0-19
Degree $2$
Conductor $1008$
Sign $0.717 - 0.696i$
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  + (−0.706 + 1.58i)3-s + (−1.27 + 0.738i)5-s + (1.34 − 2.28i)7-s + (−2.00 − 2.23i)9-s + (2.40 + 1.38i)11-s + (0.955 + 0.551i)13-s + (−0.264 − 2.54i)15-s + (1.69 − 0.978i)17-s + (3.46 − 6.00i)19-s + (2.66 + 3.73i)21-s + (0.0279 − 0.0161i)23-s + (−1.40 + 2.43i)25-s + (4.94 − 1.58i)27-s + (4.53 + 7.86i)29-s + 0.704·31-s + ⋯
L(s)  = 1  + (−0.407 + 0.913i)3-s + (−0.572 + 0.330i)5-s + (0.506 − 0.862i)7-s + (−0.667 − 0.744i)9-s + (0.724 + 0.418i)11-s + (0.264 + 0.152i)13-s + (−0.0683 − 0.657i)15-s + (0.410 − 0.237i)17-s + (0.794 − 1.37i)19-s + (0.580 + 0.814i)21-s + (0.00583 − 0.00336i)23-s + (−0.281 + 0.487i)25-s + (0.952 − 0.305i)27-s + (0.842 + 1.45i)29-s + 0.126·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.717 - 0.696i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.367082062\)
\(L(\frac12)\) \(\approx\) \(1.367082062\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.706 - 1.58i)T \)
7 \( 1 + (-1.34 + 2.28i)T \)
good5 \( 1 + (1.27 - 0.738i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.40 - 1.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.955 - 0.551i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.69 + 0.978i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.46 + 6.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0279 + 0.0161i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.53 - 7.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.704T + 31T^{2} \)
37 \( 1 + (1.92 - 3.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.23 + 1.29i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.95 + 5.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.28T + 47T^{2} \)
53 \( 1 + (-3.49 - 6.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.13T + 59T^{2} \)
61 \( 1 - 2.58iT - 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + (4.41 - 2.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 8.38iT - 79T^{2} \)
83 \( 1 + (5.34 + 9.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.83 - 5.10i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.95 + 4.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.24344777860829323696675150906, −9.271432071188008844742431150490, −8.608482565755400521617845803038, −7.29465217084368418993573753390, −6.92392225058680322738961762702, −5.57885010981083625378233018210, −4.66829266975934003409728447058, −3.95914088921023095053380762867, −3.03962049127346384497276620909, −1.01676004715372623477309775868, 0.934728685438922719344882265731, 2.13860675830766737011905513365, 3.51305262357101274195270118554, 4.71619049334149581944446864661, 5.84059034819461322311239237391, 6.20528913750611692954642453609, 7.58861761765828736597252428643, 8.087284290928658901131961151450, 8.749564250017764943505783133791, 9.842591874586168884547671000557

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