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Properties

Label 2-1008-7.3-c2-0-36
Degree $2$
Conductor $1008$
Sign $-0.476 + 0.879i$
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  + (1.27 − 0.738i)5-s + (4.08 − 5.68i)7-s + (7.76 − 13.4i)11-s − 18.6i·13-s + (−11.8 − 6.81i)17-s + (−1.20 + 0.696i)19-s + (20.0 + 34.6i)23-s + (−11.4 + 19.7i)25-s − 52.1·29-s + (−41.5 − 24.0i)31-s + (1.02 − 10.2i)35-s + (−7.93 − 13.7i)37-s + 20.6i·41-s + 16.0·43-s + (65.1 − 37.6i)47-s + ⋯
L(s)  = 1  + (0.255 − 0.147i)5-s + (0.583 − 0.811i)7-s + (0.705 − 1.22i)11-s − 1.43i·13-s + (−0.694 − 0.400i)17-s + (−0.0635 + 0.0366i)19-s + (0.870 + 1.50i)23-s + (−0.456 + 0.790i)25-s − 1.79·29-s + (−1.34 − 0.774i)31-s + (0.0293 − 0.293i)35-s + (−0.214 − 0.371i)37-s + 0.503i·41-s + 0.373·43-s + (1.38 − 0.800i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ -0.476 + 0.879i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.699404913\)
\(L(\frac12)\) \(\approx\) \(1.699404913\)
\(L(2)\) 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 \( 1 + (-4.08 + 5.68i)T \)
good5 \( 1 + (-1.27 + 0.738i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-7.76 + 13.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 18.6iT - 169T^{2} \)
17 \( 1 + (11.8 + 6.81i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.20 - 0.696i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-20.0 - 34.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 52.1T + 841T^{2} \)
31 \( 1 + (41.5 + 24.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (7.93 + 13.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 20.6iT - 1.68e3T^{2} \)
43 \( 1 - 16.0T + 1.84e3T^{2} \)
47 \( 1 + (-65.1 + 37.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (9.13 - 15.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-41.7 - 24.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23.2 + 13.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-0.310 + 0.537i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 18.7T + 5.04e3T^{2} \)
73 \( 1 + (89.5 + 51.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (55.1 + 95.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 23.8iT - 6.88e3T^{2} \)
89 \( 1 + (114. - 66.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 72.0iT - 9.40e3T^{2} \)
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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

−9.357825177103752954290361116768, −8.803004000521610256395093315913, −7.64308488293836421355266837018, −7.24294938120257609675159387061, −5.77808178711734029864828509879, −5.40245923452218139251388167235, −3.99147693719299931794193295306, −3.27485467514658239616961981321, −1.67876874860731797360117610716, −0.51957240817300006388533846855, 1.73603694574576617747006657342, 2.32749651334357679306110382853, 4.03878974108925573254452631278, 4.68312567520384744350988666426, 5.80220021892861306486499989354, 6.74628996065771828786448983493, 7.33782138530969068729048781948, 8.771776975930554897066307640259, 8.988667336828677722001146133016, 9.940789506949362571716926051925

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