↑

Properties

Label 2-656-41.36-c1-0-11
Degree $2$
Conductor $656$
Sign $0.116 + 0.993i$
Analytic cond. $5.23818$
Root an. cond. $2.28870$
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  + (−1.24 + 1.24i)3-s + (−1.15 − 0.375i)5-s + (−1.19 + 0.189i)7-s − 0.0941i·9-s + (−0.836 − 0.426i)11-s + (0.289 − 1.82i)13-s + (1.90 − 0.969i)15-s + (−1.02 + 2.01i)17-s + (−0.815 − 5.15i)19-s + (1.25 − 1.72i)21-s + (5.31 − 3.86i)23-s + (−2.85 − 2.07i)25-s + (−3.61 − 3.61i)27-s + (−0.689 − 1.35i)29-s + (0.515 + 1.58i)31-s + ⋯
L(s)  = 1  + (−0.718 + 0.718i)3-s + (−0.516 − 0.167i)5-s + (−0.451 + 0.0715i)7-s − 0.0313i·9-s + (−0.252 − 0.128i)11-s + (0.0802 − 0.506i)13-s + (0.491 − 0.250i)15-s + (−0.248 + 0.488i)17-s + (−0.187 − 1.18i)19-s + (0.272 − 0.375i)21-s + (1.10 − 0.804i)23-s + (−0.570 − 0.414i)25-s + (−0.695 − 0.695i)27-s + (−0.127 − 0.251i)29-s + (0.0926 + 0.285i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373447 - 0.332122i\)
\(L(\frac12)\) \(\approx\) \(0.373447 - 0.332122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (-4.89 + 4.13i)T \)
good3 \( 1 + (1.24 - 1.24i)T - 3iT^{2} \)
5 \( 1 + (1.15 + 0.375i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.19 - 0.189i)T + (6.65 - 2.16i)T^{2} \)
11 \( 1 + (0.836 + 0.426i)T + (6.46 + 8.89i)T^{2} \)
13 \( 1 + (-0.289 + 1.82i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.02 - 2.01i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.815 + 5.15i)T + (-18.0 + 5.87i)T^{2} \)
23 \( 1 + (-5.31 + 3.86i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.689 + 1.35i)T + (-17.0 + 23.4i)T^{2} \)
31 \( 1 + (-0.515 - 1.58i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.97 + 6.06i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (0.714 + 0.982i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-5.36 - 0.849i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (4.56 + 8.95i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-6.23 + 4.52i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.982 + 1.35i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (13.1 - 6.68i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (2.77 + 1.41i)T + (41.7 + 57.4i)T^{2} \)
73 \( 1 - 0.596iT - 73T^{2} \)
79 \( 1 + (-5.89 + 5.89i)T - 79iT^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (-6.08 + 0.964i)T + (84.6 - 27.5i)T^{2} \)
97 \( 1 + (15.0 - 7.65i)T + (57.0 - 78.4i)T^{2} \)
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

−10.59652371824534401145095686082, −9.571246747189168681899356736833, −8.686568429795463519014564067336, −7.74590174403719673623636591323, −6.66194657338396408687810736820, −5.67471901996524332795936220854, −4.79349489668846291263417244175, −3.94098676935667102618576577187, −2.60430754014514247845507131687, −0.30955481141753725622364061039, 1.38658269818172919742894953134, 3.08032933235181821822195963807, 4.21316294200174950599548268600, 5.50990017768795556839359379487, 6.34955026087309240933687678549, 7.16752186528265931881637414321, 7.83371342718933692165148951057, 9.085534223815999996656591265803, 9.871942148134934092739367741824, 11.01737780800722200194418700963

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