↑

Properties

Label 2-418-209.94-c1-0-14
Degree $2$
Conductor $418$
Sign $-0.543 + 0.839i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
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  + (−0.309 − 0.951i)2-s + (0.775 − 1.06i)3-s + (−0.809 + 0.587i)4-s + (1.01 − 3.11i)5-s + (−1.25 − 0.407i)6-s + (1.17 + 1.61i)7-s + (0.809 + 0.587i)8-s + (0.388 + 1.19i)9-s − 3.27·10-s + (1.22 − 3.08i)11-s + 1.31i·12-s + (−1.74 − 5.35i)13-s + (1.17 − 1.61i)14-s + (−2.54 − 3.49i)15-s + (0.309 − 0.951i)16-s + (5.89 + 1.91i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.447 − 0.616i)3-s + (−0.404 + 0.293i)4-s + (0.452 − 1.39i)5-s + (−0.512 − 0.166i)6-s + (0.443 + 0.610i)7-s + (0.286 + 0.207i)8-s + (0.129 + 0.398i)9-s − 1.03·10-s + (0.368 − 0.929i)11-s + 0.381i·12-s + (−0.482 − 1.48i)13-s + (0.313 − 0.431i)14-s + (−0.656 − 0.903i)15-s + (0.0772 − 0.237i)16-s + (1.42 + 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.730793 - 1.34343i\)
\(L(\frac12)\) \(\approx\) \(0.730793 - 1.34343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-1.22 + 3.08i)T \)
19 \( 1 + (3.26 - 2.88i)T \)
good3 \( 1 + (-0.775 + 1.06i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.01 + 3.11i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.17 - 1.61i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.74 + 5.35i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-5.89 - 1.91i)T + (13.7 + 9.99i)T^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 + (6.67 - 4.85i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.26 - 0.410i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.96 + 6.82i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.46 - 6.15i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.02iT - 43T^{2} \)
47 \( 1 + (0.734 + 0.533i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.37 + 1.42i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.71 + 2.35i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-10.2 - 3.33i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.93iT - 67T^{2} \)
71 \( 1 + (-1.56 - 0.509i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.91 + 8.13i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.75 - 14.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-12.1 - 3.95i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.44iT - 89T^{2} \)
97 \( 1 + (9.41 - 3.05i)T + (78.4 - 57.0i)T^{2} \)
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

−10.84235213035482596481158461638, −9.963076020109163961411407025417, −8.934078232030532320725859613844, −8.224024521811723775148085352419, −7.76683800187578271423085040291, −5.74098766690239048995498561155, −5.23155302142198104046393389357, −3.66681908845292118987971611188, −2.19060992107572028150145171246, −1.13804001357164739939452575780, 2.12036884225158682907846543716, 3.72755478294407284944260596794, 4.55873813151811735192888982929, 6.09059275253245962417712081652, 7.02732351663165440841801066268, 7.49620354064388711239092264658, 8.972862411753815593933609534437, 9.806395561630211164034253283611, 10.17604935584546994176117556564, 11.31158447011385817914039746680

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