↑

Properties

Label 2-418-209.151-c1-0-15
Degree $2$
Conductor $418$
Sign $0.581 + 0.813i$
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  + (0.809 − 0.587i)2-s + (2.42 − 0.787i)3-s + (0.309 − 0.951i)4-s + (1.61 + 1.17i)5-s + (1.49 − 2.06i)6-s + (−2.37 − 0.770i)7-s + (−0.309 − 0.951i)8-s + (2.82 − 2.05i)9-s + 1.99·10-s + (−2.62 − 2.02i)11-s − 2.54i·12-s + (0.306 − 0.222i)13-s + (−2.37 + 0.770i)14-s + (4.83 + 1.57i)15-s + (−0.809 − 0.587i)16-s + (−1.98 + 2.72i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (1.39 − 0.454i)3-s + (0.154 − 0.475i)4-s + (0.722 + 0.525i)5-s + (0.611 − 0.841i)6-s + (−0.896 − 0.291i)7-s + (−0.109 − 0.336i)8-s + (0.941 − 0.684i)9-s + 0.631·10-s + (−0.792 − 0.610i)11-s − 0.735i·12-s + (0.0851 − 0.0618i)13-s + (−0.634 + 0.206i)14-s + (1.24 + 0.406i)15-s + (−0.202 − 0.146i)16-s + (−0.480 + 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50428 - 1.28742i\)
\(L(\frac12)\) \(\approx\) \(2.50428 - 1.28742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (2.62 + 2.02i)T \)
19 \( 1 + (-1.14 - 4.20i)T \)
good3 \( 1 + (-2.42 + 0.787i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (2.37 + 0.770i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.306 + 0.222i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.98 - 2.72i)T + (-5.25 - 16.1i)T^{2} \)
23 \( 1 - 9.33T + 23T^{2} \)
29 \( 1 + (2.55 - 7.87i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.17 - 4.37i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (9.81 + 3.18i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.321 + 0.989i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.69iT - 43T^{2} \)
47 \( 1 + (3.57 + 11.0i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.76 - 5.18i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.80 + 1.56i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.57 + 3.53i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.35iT - 67T^{2} \)
71 \( 1 + (-0.523 + 0.720i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.9 + 3.88i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.93 + 7.21i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.00 + 4.14i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.43iT - 89T^{2} \)
97 \( 1 + (-7.02 - 9.66i)T + (-29.9 + 92.2i)T^{2} \)
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

−10.77082496860148661069734750742, −10.31698766766288891250072124752, −9.228715728388503638348181653097, −8.471799726047605034335336080208, −7.22337884066211206020584405941, −6.45862100182439846980749795347, −5.26515734107012826736693938335, −3.44609877281370844525800996169, −3.04209614391620350423199609255, −1.78902488264240272270342479965, 2.40030073021782909445710306131, 3.13269759759446054164486408668, 4.51388342843301151171787747257, 5.39627980789773862947895173561, 6.71485228493883341972345337869, 7.65303697838426572138460618052, 8.823635576010412556900772559864, 9.326192097670347394679949706880, 10.02453185780216556867855289123, 11.41018410377522301766295008465

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