↑

Properties

Label 2-208-13.2-c2-0-3
Degree $2$
Conductor $208$
Sign $-0.892 - 0.451i$
Analytic cond. $5.66758$
Root an. cond. $2.38066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.10 + 3.64i)3-s + (−5.99 + 5.99i)5-s + (0.737 − 0.197i)7-s + (−4.34 + 7.52i)9-s + (2.43 − 9.09i)11-s + (−10.6 + 7.38i)13-s + (−34.4 − 9.22i)15-s + (−10.3 − 5.98i)17-s + (7.11 + 26.5i)19-s + (2.26 + 2.26i)21-s + (20.1 − 11.6i)23-s − 46.8i·25-s + 1.28·27-s + (0.329 + 0.570i)29-s + (−33.8 + 33.8i)31-s + ⋯
L(s)  = 1  + (0.701 + 1.21i)3-s + (−1.19 + 1.19i)5-s + (0.105 − 0.0282i)7-s + (−0.482 + 0.836i)9-s + (0.221 − 0.826i)11-s + (−0.822 + 0.568i)13-s + (−2.29 − 0.615i)15-s + (−0.610 − 0.352i)17-s + (0.374 + 1.39i)19-s + (0.108 + 0.108i)21-s + (0.878 − 0.506i)23-s − 1.87i·25-s + 0.0476·27-s + (0.0113 + 0.0196i)29-s + (−1.09 + 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(5.66758\)
Root analytic conductor: \(2.38066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1),\ -0.892 - 0.451i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.297837 + 1.24850i\)
\(L(\frac12)\) \(\approx\) \(0.297837 + 1.24850i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (10.6 - 7.38i)T \)
good3 \( 1 + (-2.10 - 3.64i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (5.99 - 5.99i)T - 25iT^{2} \)
7 \( 1 + (-0.737 + 0.197i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-2.43 + 9.09i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (10.3 + 5.98i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-7.11 - 26.5i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-20.1 + 11.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.329 - 0.570i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (33.8 - 33.8i)T - 961iT^{2} \)
37 \( 1 + (18.6 - 69.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-65.3 - 17.5i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-23.1 - 13.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-13.2 - 13.2i)T + 2.20e3iT^{2} \)
53 \( 1 - 15.5T + 2.80e3T^{2} \)
59 \( 1 + (-59.8 + 16.0i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-36.0 + 62.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (36.7 + 9.83i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (21.7 + 81.1i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (5.13 + 5.13i)T + 5.32e3iT^{2} \)
79 \( 1 - 114.T + 6.24e3T^{2} \)
83 \( 1 + (-12.9 + 12.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (19.6 - 73.4i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (15.8 + 59.2i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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

−12.33415860042999465631526775132, −11.29677587062492680348438892692, −10.69268245200531175479355225108, −9.682862349942126171145592869554, −8.658357450873664867502961023138, −7.66637483487310620935316447872, −6.58278346439400967242618541335, −4.75364656962552303844490966358, −3.69641726936657647994237292537, −2.92770555478108415523083657960, 0.67269783799680872718287929774, 2.31295098593715842410295547053, 4.06328940197514847283736730447, 5.24604291302648706790491652169, 7.32044315144005248521821412185, 7.40787115174776683765432536173, 8.665961946892572430631125208483, 9.290460547001728425252813972078, 11.10716737845427674083505794927, 12.07386323437063159488086684977

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