↑

Properties

Label 2-208-16.13-c1-0-5
Degree $2$
Conductor $208$
Sign $-0.245 - 0.969i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
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  + (1.14 + 0.827i)2-s + (−0.390 − 0.390i)3-s + (0.630 + 1.89i)4-s + (−2.58 + 2.58i)5-s + (−0.124 − 0.771i)6-s + 2.97i·7-s + (−0.848 + 2.69i)8-s − 2.69i·9-s + (−5.10 + 0.824i)10-s + (2.25 − 2.25i)11-s + (0.495 − 0.988i)12-s + (−0.707 − 0.707i)13-s + (−2.46 + 3.41i)14-s + 2.02·15-s + (−3.20 + 2.39i)16-s + 6.77·17-s + ⋯
L(s)  = 1  + (0.810 + 0.585i)2-s + (−0.225 − 0.225i)3-s + (0.315 + 0.949i)4-s + (−1.15 + 1.15i)5-s + (−0.0509 − 0.315i)6-s + 1.12i·7-s + (−0.299 + 0.953i)8-s − 0.898i·9-s + (−1.61 + 0.260i)10-s + (0.679 − 0.679i)11-s + (0.143 − 0.285i)12-s + (−0.196 − 0.196i)13-s + (−0.658 + 0.912i)14-s + 0.521·15-s + (−0.801 + 0.598i)16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.245 - 0.969i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ -0.245 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.896480 + 1.15218i\)
\(L(\frac12)\) \(\approx\) \(0.896480 + 1.15218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 0.827i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.390 + 0.390i)T + 3iT^{2} \)
5 \( 1 + (2.58 - 2.58i)T - 5iT^{2} \)
7 \( 1 - 2.97iT - 7T^{2} \)
11 \( 1 + (-2.25 + 2.25i)T - 11iT^{2} \)
17 \( 1 - 6.77T + 17T^{2} \)
19 \( 1 + (-3.46 - 3.46i)T + 19iT^{2} \)
23 \( 1 - 1.42iT - 23T^{2} \)
29 \( 1 + (6.93 + 6.93i)T + 29iT^{2} \)
31 \( 1 - 8.57T + 31T^{2} \)
37 \( 1 + (-1.68 + 1.68i)T - 37iT^{2} \)
41 \( 1 + 4.08iT - 41T^{2} \)
43 \( 1 + (4.98 - 4.98i)T - 43iT^{2} \)
47 \( 1 + 5.83T + 47T^{2} \)
53 \( 1 + (-3.07 + 3.07i)T - 53iT^{2} \)
59 \( 1 + (4.22 - 4.22i)T - 59iT^{2} \)
61 \( 1 + (0.0334 + 0.0334i)T + 61iT^{2} \)
67 \( 1 + (-4.75 - 4.75i)T + 67iT^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 - 2.39iT - 73T^{2} \)
79 \( 1 - 7.39T + 79T^{2} \)
83 \( 1 + (-2.60 - 2.60i)T + 83iT^{2} \)
89 \( 1 - 1.52iT - 89T^{2} \)
97 \( 1 + 5.49T + 97T^{2} \)
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

−12.31623104105219694376289845801, −11.88191754227427541400237106104, −11.36260201354446470274012727469, −9.681998686913535947348311927352, −8.238989366943948710936781618182, −7.49369058228284415926179408178, −6.34839034489127617492289862235, −5.64187388297497769585875690342, −3.76452472029239221620745238698, −3.07411338785148239707754875691, 1.15701842685214432811679308182, 3.55884935420307349196861811556, 4.55435579905231872589241373354, 5.17385787457652878298295069827, 7.03591055290229760088069361629, 7.923909417710130092071655989615, 9.450136021942709152345060049975, 10.36321863974143807558052574188, 11.44025123619291492415478068958, 12.04325856182085071275006203839

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