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Properties

Label 2-1008-84.83-c2-0-18
Degree $2$
Conductor $1008$
Sign $0.998 + 0.0627i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.41·5-s + (3.32 − 6.16i)7-s − 17.6·11-s + 17.9i·13-s + 13.3·17-s + 14.6·19-s + 30.1·23-s + 63.6·25-s + 25.9i·29-s + 11.5·31-s + (31.2 − 57.9i)35-s − 17.9·37-s + 12.6·41-s − 19.2i·43-s + 48.3i·47-s + ⋯
L(s)  = 1  + 1.88·5-s + (0.474 − 0.880i)7-s − 1.60·11-s + 1.37i·13-s + 0.787·17-s + 0.770·19-s + 1.31·23-s + 2.54·25-s + 0.894i·29-s + 0.372·31-s + (0.893 − 1.65i)35-s − 0.484·37-s + 0.308·41-s − 0.447i·43-s + 1.02i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.998 + 0.0627i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.998 + 0.0627i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.879903522\)
\(L(\frac12)\) \(\approx\) \(2.879903522\)
\(L(2)\) 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 \( 1 + (-3.32 + 6.16i)T \)
good5 \( 1 - 9.41T + 25T^{2} \)
11 \( 1 + 17.6T + 121T^{2} \)
13 \( 1 - 17.9iT - 169T^{2} \)
17 \( 1 - 13.3T + 289T^{2} \)
19 \( 1 - 14.6T + 361T^{2} \)
23 \( 1 - 30.1T + 529T^{2} \)
29 \( 1 - 25.9iT - 841T^{2} \)
31 \( 1 - 11.5T + 961T^{2} \)
37 \( 1 + 17.9T + 1.36e3T^{2} \)
41 \( 1 - 12.6T + 1.68e3T^{2} \)
43 \( 1 + 19.2iT - 1.84e3T^{2} \)
47 \( 1 - 48.3iT - 2.20e3T^{2} \)
53 \( 1 - 3.42iT - 2.80e3T^{2} \)
59 \( 1 + 53.9iT - 3.48e3T^{2} \)
61 \( 1 + 59.8iT - 3.72e3T^{2} \)
67 \( 1 + 104. iT - 4.48e3T^{2} \)
71 \( 1 - 76.5T + 5.04e3T^{2} \)
73 \( 1 + 142. iT - 5.32e3T^{2} \)
79 \( 1 - 47.9iT - 6.24e3T^{2} \)
83 \( 1 - 90.9iT - 6.88e3T^{2} \)
89 \( 1 - 122.T + 7.92e3T^{2} \)
97 \( 1 - 94.7iT - 9.40e3T^{2} \)
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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

−9.679057529041903291401507732101, −9.245034173646482776178743688580, −8.061640990777345715339737621077, −7.14353329313766964434738548143, −6.38579742838214270486645376150, −5.20248099534965453682179029661, −4.93662141794603995316420891968, −3.23519748824984123705204780523, −2.13601275316639120826354912845, −1.16557799424984176785176527876, 1.11790569346977559910830555625, 2.48350288322211140121183946589, 2.91400519083219295474163137350, 5.13517892815040565088712260868, 5.37018492348337696501261939376, 6.00745564768054008005881484476, 7.32092568058666167916391277320, 8.213361394316939122123845840093, 8.998915320620387616687135457298, 10.02868678817087524360968621316

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