↑

Properties

Label 2-208-16.13-c1-0-16
Degree $2$
Conductor $208$
Sign $-0.196 + 0.980i$
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  + (−0.0687 − 1.41i)2-s + (0.559 + 0.559i)3-s + (−1.99 + 0.194i)4-s + (0.101 − 0.101i)5-s + (0.751 − 0.828i)6-s − 3.07i·7-s + (0.411 + 2.79i)8-s − 2.37i·9-s + (−0.150 − 0.136i)10-s + (3.69 − 3.69i)11-s + (−1.22 − 1.00i)12-s + (−0.707 − 0.707i)13-s + (−4.34 + 0.211i)14-s + 0.113·15-s + (3.92 − 0.773i)16-s − 3.08·17-s + ⋯
L(s)  = 1  + (−0.0486 − 0.998i)2-s + (0.322 + 0.322i)3-s + (−0.995 + 0.0970i)4-s + (0.0455 − 0.0455i)5-s + (0.306 − 0.338i)6-s − 1.16i·7-s + (0.145 + 0.989i)8-s − 0.791i·9-s + (−0.0477 − 0.0432i)10-s + (1.11 − 1.11i)11-s + (−0.352 − 0.289i)12-s + (−0.196 − 0.196i)13-s + (−1.16 + 0.0565i)14-s + 0.0294·15-s + (0.981 − 0.193i)16-s − 0.748·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.196 + 0.980i$
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: \((2,\ 208,\ (\ :1/2),\ -0.196 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752602 - 0.918776i\)
\(L(\frac12)\) \(\approx\) \(0.752602 - 0.918776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0687 + 1.41i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.559 - 0.559i)T + 3iT^{2} \)
5 \( 1 + (-0.101 + 0.101i)T - 5iT^{2} \)
7 \( 1 + 3.07iT - 7T^{2} \)
11 \( 1 + (-3.69 + 3.69i)T - 11iT^{2} \)
17 \( 1 + 3.08T + 17T^{2} \)
19 \( 1 + (-4.18 - 4.18i)T + 19iT^{2} \)
23 \( 1 - 6.17iT - 23T^{2} \)
29 \( 1 + (2.52 + 2.52i)T + 29iT^{2} \)
31 \( 1 + 3.13T + 31T^{2} \)
37 \( 1 + (7.65 - 7.65i)T - 37iT^{2} \)
41 \( 1 - 2.83iT - 41T^{2} \)
43 \( 1 + (-1.76 + 1.76i)T - 43iT^{2} \)
47 \( 1 - 9.54T + 47T^{2} \)
53 \( 1 + (-5.63 + 5.63i)T - 53iT^{2} \)
59 \( 1 + (-5.74 + 5.74i)T - 59iT^{2} \)
61 \( 1 + (2.87 + 2.87i)T + 61iT^{2} \)
67 \( 1 + (-9.62 - 9.62i)T + 67iT^{2} \)
71 \( 1 + 2.52iT - 71T^{2} \)
73 \( 1 - 17.0iT - 73T^{2} \)
79 \( 1 + 9.09T + 79T^{2} \)
83 \( 1 + (4.93 + 4.93i)T + 83iT^{2} \)
89 \( 1 - 0.851iT - 89T^{2} \)
97 \( 1 - 5.16T + 97T^{2} \)
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

−11.82834552005969436903454659961, −11.26425314453855000950472606430, −10.10184916548510486346878396483, −9.393479769263942481758175907135, −8.488257127307428630962139417061, −7.14210198032915578611145143573, −5.58768888885047798553370204563, −3.90463084571399640161016079356, −3.46778308184032981976928355218, −1.17794560991673847247446215151, 2.26934884028239130009842307106, 4.36111977507826497753237872479, 5.41533628615499994936449241571, 6.72125019934865649116277310301, 7.43944527435613688517886648543, 8.847280045153693542612914980692, 9.127095979938701958887207435500, 10.53729126320603199750108389785, 12.05025449322367600091403411145, 12.71660418474232084407689795580

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