↑

Properties

Label 2-208-13.10-c1-0-4
Degree $2$
Conductor $208$
Sign $0.398 + 0.917i$
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  + (0.193 + 0.334i)3-s − 3.17i·5-s + (−2.98 − 1.72i)7-s + (1.42 − 2.46i)9-s + (2.98 − 1.72i)11-s + (−0.362 + 3.58i)13-s + (1.06 − 0.613i)15-s + (−0.886 + 1.53i)17-s + (2.88 + 1.66i)19-s − 1.33i·21-s + (0.193 + 0.334i)23-s − 5.07·25-s + 2.26·27-s + (2.28 + 3.96i)29-s − 11.0i·31-s + ⋯
L(s)  = 1  + (0.111 + 0.193i)3-s − 1.41i·5-s + (−1.12 − 0.650i)7-s + (0.475 − 0.822i)9-s + (0.898 − 0.518i)11-s + (−0.100 + 0.994i)13-s + (0.274 − 0.158i)15-s + (−0.215 + 0.372i)17-s + (0.661 + 0.382i)19-s − 0.290i·21-s + (0.0402 + 0.0697i)23-s − 1.01·25-s + 0.435·27-s + (0.424 + 0.735i)29-s − 1.98i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.398 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.962600 - 0.631208i\)
\(L(\frac12)\) \(\approx\) \(0.962600 - 0.631208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.362 - 3.58i)T \)
good3 \( 1 + (-0.193 - 0.334i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.17iT - 5T^{2} \)
7 \( 1 + (2.98 + 1.72i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.98 + 1.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.886 - 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.88 - 1.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.193 - 0.334i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.28 - 3.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 11.0iT - 31T^{2} \)
37 \( 1 + (1.40 - 0.810i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.96 - 1.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.36 - 9.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.11iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (-5.47 - 3.16i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.21 - 2.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.91 + 4.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.88 - 5.12i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.40iT - 73T^{2} \)
79 \( 1 + 8.22T + 79T^{2} \)
83 \( 1 + 1.11iT - 83T^{2} \)
89 \( 1 + (15.4 - 8.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.88 - 2.24i)T + (48.5 + 84.0i)T^{2} \)
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

−12.32765414326599675911928031798, −11.44749990227751836661872392890, −9.806634825764168657283794710602, −9.413059907123763526516593409649, −8.477569518940819106645149523066, −6.97177017377193432678477924863, −6.04733843940162153577053456352, −4.43548036402817540838517116076, −3.63697916152885676023342731551, −1.09152126373637792741777817710, 2.45405001847680790829964400016, 3.48177403081796710384908377767, 5.33106156504772528924087079571, 6.75427000183872454866379814863, 7.11629853739326235304258856928, 8.609835040578601814292960492275, 9.913236358691717243486011498882, 10.40809460914211047462076511367, 11.64709163484843525886256330883, 12.55479456768813631251895898461

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