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Properties

Label 2-1008-63.59-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.895 + 0.444i$
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  + (−0.994 − 1.41i)3-s − 1.58·5-s + (1.06 − 2.42i)7-s + (−1.02 + 2.81i)9-s + 3.95i·11-s + (5.09 + 2.93i)13-s + (1.57 + 2.24i)15-s + (−3.19 + 5.53i)17-s + (6.37 − 3.68i)19-s + (−4.49 + 0.898i)21-s − 2.43i·23-s − 2.48·25-s + (5.01 − 1.35i)27-s + (4.34 − 2.50i)29-s + (−0.855 + 0.493i)31-s + ⋯
L(s)  = 1  + (−0.573 − 0.818i)3-s − 0.709·5-s + (0.402 − 0.915i)7-s + (−0.341 + 0.939i)9-s + 1.19i·11-s + (1.41 + 0.815i)13-s + (0.407 + 0.580i)15-s + (−0.775 + 1.34i)17-s + (1.46 − 0.844i)19-s + (−0.980 + 0.196i)21-s − 0.506i·23-s − 0.496·25-s + (0.965 − 0.260i)27-s + (0.806 − 0.465i)29-s + (−0.153 + 0.0886i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.216944009\)
\(L(\frac12)\) \(\approx\) \(1.216944009\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.994 + 1.41i)T \)
7 \( 1 + (-1.06 + 2.42i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 - 3.95iT - 11T^{2} \)
13 \( 1 + (-5.09 - 2.93i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.19 - 5.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.37 + 3.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.43iT - 23T^{2} \)
29 \( 1 + (-4.34 + 2.50i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.855 - 0.493i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.183 - 0.317i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.58 + 9.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.26 + 2.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.543 + 0.940i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.89 - 3.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.73 - 4.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.4 - 7.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.70 + 9.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.65iT - 71T^{2} \)
73 \( 1 + (-7.64 - 4.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.30 - 9.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.96 + 5.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.70 - 8.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.4 + 6.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15450102918832640705945201565, −8.843299147759764137732687748902, −8.099538035540563022584796441867, −7.19702083626047162063619263511, −6.81082874766683326279832370233, −5.70572210051697617723007865451, −4.48000295366775237280606810711, −3.90279548324459464714152024017, −2.11004320145430812804325071799, −0.969913276830428084512010135223, 0.865005427821782577566994347812, 3.04073583898555267754880926855, 3.66518796843746505915405584751, 4.92844714230243631886405059231, 5.63880453426916499862669590408, 6.31962300209262611416549304089, 7.72809157710831565907660629450, 8.468781016100802976135324896195, 9.156394496485312139154631862013, 10.04567821600096437981290222908

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