↑

Properties

Label 2-616-77.41-c1-0-21
Degree $2$
Conductor $616$
Sign $0.646 + 0.763i$
Analytic cond. $4.91878$
Root an. cond. $2.21783$
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  + (1.74 − 0.565i)3-s + (2.00 − 2.75i)5-s + (2.13 − 1.56i)7-s + (0.282 − 0.205i)9-s + (1.95 + 2.67i)11-s + (−3.12 + 2.26i)13-s + (1.92 − 5.92i)15-s + (5.18 + 3.76i)17-s + (−1.35 − 4.18i)19-s + (2.83 − 3.92i)21-s − 8.95·23-s + (−2.03 − 6.27i)25-s + (−2.85 + 3.92i)27-s + (−2.95 − 0.961i)29-s + (−0.176 − 0.242i)31-s + ⋯
L(s)  = 1  + (1.00 − 0.326i)3-s + (0.895 − 1.23i)5-s + (0.807 − 0.590i)7-s + (0.0942 − 0.0684i)9-s + (0.589 + 0.807i)11-s + (−0.865 + 0.629i)13-s + (0.497 − 1.53i)15-s + (1.25 + 0.913i)17-s + (−0.311 − 0.959i)19-s + (0.618 − 0.856i)21-s − 1.86·23-s + (−0.407 − 1.25i)25-s + (−0.548 + 0.755i)27-s + (−0.549 − 0.178i)29-s + (−0.0316 − 0.0436i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $0.646 + 0.763i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ 0.646 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22095 - 1.02992i\)
\(L(\frac12)\) \(\approx\) \(2.22095 - 1.02992i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.13 + 1.56i)T \)
11 \( 1 + (-1.95 - 2.67i)T \)
good3 \( 1 + (-1.74 + 0.565i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.00 + 2.75i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.12 - 2.26i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.18 - 3.76i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.35 + 4.18i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.95T + 23T^{2} \)
29 \( 1 + (2.95 + 0.961i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.176 + 0.242i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.331 + 1.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.131 - 0.404i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.11iT - 43T^{2} \)
47 \( 1 + (2.54 - 0.827i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-10.0 + 7.30i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.66 + 1.18i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.45 - 6.14i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 + (12.9 + 9.37i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.20 - 6.77i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.31 - 5.94i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.928 + 0.674i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.58iT - 89T^{2} \)
97 \( 1 + (1.20 + 1.65i)T + (-29.9 + 92.2i)T^{2} \)
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

−10.10628694817537808631023462740, −9.588151129042187082760964612581, −8.706613222226726535144721591924, −8.011948849369435795837251861778, −7.22056962723936811426904177886, −5.87416383749755014459238773427, −4.82914366940291664991842104433, −3.99527891931610275652458765780, −2.18345622779691534716984027969, −1.51819912626969854230709132097, 2.03345409597298562631132211936, 2.88916095704423994789125952446, 3.79575808576348509777930088690, 5.49853462174348065292433864992, 6.04732942122309693990985521909, 7.42299266329557992300039935456, 8.140601228039087312642584442179, 9.068118442068037612224528390108, 9.944404554082379375194516988320, 10.40403839362428562366955761120

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