↑

Properties

Label 2-418-11.9-c1-0-6
Degree $2$
Conductor $418$
Sign $0.999 + 0.00569i$
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 + (−2.04 + 1.48i)3-s + (−0.809 − 0.587i)4-s + (0.0815 + 0.251i)5-s + (−0.781 − 2.40i)6-s + (−3.52 − 2.55i)7-s + (0.809 − 0.587i)8-s + (1.05 − 3.23i)9-s − 0.264·10-s + (3.09 + 1.18i)11-s + 2.53·12-s + (1.77 − 5.46i)13-s + (3.52 − 2.55i)14-s + (−0.540 − 0.392i)15-s + (0.309 + 0.951i)16-s + (1.40 + 4.33i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.18 + 0.858i)3-s + (−0.404 − 0.293i)4-s + (0.0364 + 0.112i)5-s + (−0.319 − 0.982i)6-s + (−1.33 − 0.967i)7-s + (0.286 − 0.207i)8-s + (0.350 − 1.07i)9-s − 0.0835·10-s + (0.933 + 0.357i)11-s + 0.730·12-s + (0.492 − 1.51i)13-s + (0.941 − 0.683i)14-s + (−0.139 − 0.101i)15-s + (0.0772 + 0.237i)16-s + (0.341 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.999 + 0.00569i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.999 + 0.00569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.563427 - 0.00160543i\)
\(L(\frac12)\) \(\approx\) \(0.563427 - 0.00160543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.09 - 1.18i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (2.04 - 1.48i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.0815 - 0.251i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.52 + 2.55i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.77 + 5.46i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.40 - 4.33i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + 6.87T + 23T^{2} \)
29 \( 1 + (-4.05 - 2.94i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.94 + 5.97i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.25 + 2.36i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.505 + 0.367i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (-8.36 + 6.07i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.34 + 13.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.27 + 0.927i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.56 + 4.82i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 + (0.833 + 2.56i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.24 + 3.81i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.51 - 4.67i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.73 + 8.41i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 7.63T + 89T^{2} \)
97 \( 1 + (-3.40 + 10.4i)T + (-78.4 - 57.0i)T^{2} \)
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

−10.70717536213895958671760799587, −10.31106382138485309523849365866, −9.751742293279074937519328776300, −8.475553027011856440659155918876, −7.24750116117333738001183854714, −6.18098936162009651734193879637, −5.85623254596831955472181276708, −4.35233256633177784736102448871, −3.61781464574895878479682098877, −0.53518069617203066337701031291, 1.21212218661377515893434319282, 2.76688186896465726775250943602, 4.26005943144093186050011003603, 5.77016436757469622161473899682, 6.38176060912460498174838592857, 7.20704705190047545720612406390, 8.906548784355788421755075640286, 9.269886600160105183141395217344, 10.48426488657886115934731678930, 11.55570857036497765202948110481

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