↑

Properties

Label 2-656-41.37-c1-0-13
Degree $2$
Conductor $656$
Sign $0.999 + 0.0354i$
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  + 2.90·3-s + (0.0357 − 0.109i)5-s + (1.92 + 1.39i)7-s + 5.45·9-s + (−1.21 − 3.73i)11-s + (1.37 − 0.998i)13-s + (0.103 − 0.319i)15-s + (0.473 + 1.45i)17-s + (−5.61 − 4.07i)19-s + (5.58 + 4.05i)21-s + (−3.68 + 2.67i)23-s + (4.03 + 2.93i)25-s + 7.14·27-s + (−2.72 + 8.37i)29-s + (0.699 + 2.15i)31-s + ⋯
L(s)  = 1  + 1.67·3-s + (0.0159 − 0.0491i)5-s + (0.725 + 0.527i)7-s + 1.81·9-s + (−0.365 − 1.12i)11-s + (0.381 − 0.276i)13-s + (0.0268 − 0.0825i)15-s + (0.114 + 0.353i)17-s + (−1.28 − 0.935i)19-s + (1.21 + 0.885i)21-s + (−0.768 + 0.558i)23-s + (0.806 + 0.586i)25-s + 1.37·27-s + (−0.505 + 1.55i)29-s + (0.125 + 0.386i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.999 + 0.0354i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.999 + 0.0354i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72144 - 0.0482563i\)
\(L(\frac12)\) \(\approx\) \(2.72144 - 0.0482563i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.19 + 3.74i)T \)
good3 \( 1 - 2.90T + 3T^{2} \)
5 \( 1 + (-0.0357 + 0.109i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.92 - 1.39i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (1.21 + 3.73i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.37 + 0.998i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.473 - 1.45i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.61 + 4.07i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.68 - 2.67i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.72 - 8.37i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.699 - 2.15i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.395 + 1.21i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-4.73 + 3.44i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (7.04 - 5.11i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.00 - 6.17i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.62 - 1.17i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (6.64 + 4.83i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-0.327 + 1.00i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.57 + 14.0i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 4.43T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + (-4.31 - 3.13i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.49 - 4.60i)T + (-78.4 - 57.0i)T^{2} \)
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

−10.61019097679084541306827689283, −9.205771243253788732582333366050, −8.783751776663717383594675282548, −8.162836077762478607012916356506, −7.36398633709621500387877665047, −6.05552466765009915821009773333, −4.87889210634341416194861281420, −3.64909174454466041562434186254, −2.80624881761955748358770833744, −1.67371258755362721194115106498, 1.76919362130062796868993771677, 2.60457841710134745768513810364, 4.06937298489871163874358774535, 4.50145846298000795361329062917, 6.26037606113374104801974922529, 7.40491036631595701326165600319, 8.019856834358834672845090340361, 8.602274340868278095737199433485, 9.732362548229060110605662198644, 10.19030225579678643636654999584

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