↑

Properties

Label 2-1008-63.4-c1-0-45
Degree $2$
Conductor $1008$
Sign $-0.993 - 0.111i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.21 − 1.22i)3-s − 0.481·5-s + (−2.53 + 0.763i)7-s + (−0.0248 − 2.99i)9-s − 3.38·11-s + (−2.86 + 4.95i)13-s + (−0.587 + 0.592i)15-s + (2.75 − 4.77i)17-s + (−2.18 − 3.77i)19-s + (−2.15 + 4.04i)21-s − 3.62·23-s − 4.76·25-s + (−3.71 − 3.62i)27-s + (1.53 + 2.65i)29-s + (−4.67 − 8.09i)31-s + ⋯
L(s)  = 1  + (0.704 − 0.710i)3-s − 0.215·5-s + (−0.957 + 0.288i)7-s + (−0.00827 − 0.999i)9-s − 1.01·11-s + (−0.793 + 1.37i)13-s + (−0.151 + 0.152i)15-s + (0.668 − 1.15i)17-s + (−0.500 − 0.866i)19-s + (−0.469 + 0.882i)21-s − 0.756·23-s − 0.953·25-s + (−0.715 − 0.698i)27-s + (0.284 + 0.492i)29-s + (−0.839 − 1.45i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.993 - 0.111i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3751826741\)
\(L(\frac12)\) \(\approx\) \(0.3751826741\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.21 + 1.22i)T \)
7 \( 1 + (2.53 - 0.763i)T \)
good5 \( 1 + 0.481T + 5T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 + (2.86 - 4.95i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.18 + 3.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.62T + 23T^{2} \)
29 \( 1 + (-1.53 - 2.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.67 + 8.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.48 - 2.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.29 - 10.9i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 + 3.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.88 - 3.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.57 + 9.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.21 - 7.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.64 + 6.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.28 - 2.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.94T + 71T^{2} \)
73 \( 1 + (0.862 - 1.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.79 - 4.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.119 - 0.206i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.648 - 1.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.02 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.650145902024647667794243032358, −8.687537255104356272220435681801, −7.78410705458730690999943978681, −7.07625456530581939704957166853, −6.38063677340077149893663573613, −5.21656830239996237240097981610, −3.99796204186463058392775703880, −2.86944416807700114794077933387, −2.13401459746356845040826833840, −0.14071003006380334100682813213, 2.21899443947332575790333029110, 3.33317331184022482440201587227, 3.90802329904176784246122840356, 5.23963078414926381278367163518, 5.91720205733476111813542362460, 7.35728119789881355823751372686, 7.966860585998899530020979636190, 8.639087763977474700622962857906, 9.877108594862990881271920524683, 10.26185510763934607644673840682

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