↑

Properties

Label 1-197-197.196-r0-0-0
Degree $1$
Conductor $197$
Sign $1$
Analytic cond. $0.914864$
Root an. cond. $0.914864$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $1$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{197} (196, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4332289955\)
\(L(\frac12)\) \(\approx\) \(0.4332289955\)
\(L(1)\) \(\approx\) \(0.4750008885\)
\(L(1)\) \(\approx\) \(0.4750008885\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.91015549878388112275243373098, −26.69179915172111168984860069084, −24.8797145776674866369859770361, −24.10096993852439686588833832324, −23.51964211783655315769254609814, −22.20091746193614388946818149208, −21.11312455363948113257821184828, −20.161410190655384488370337217570, −19.101041912396701593884952390017, −18.11625822723174927712180828683, −17.551313996657898695121121330989, −16.41218087657527046400338574975, −15.658543266406915149414498187174, −14.78558369990380964076016332804, −12.793928035823971873550369159217, −11.77103220035168417289622195039, −11.1348867056033258260813644738, −10.32794976444752957642728447190, −8.90405576737827996770584228794, −7.60149056272632292052777748313, −7.18845902309268000133166862156, −5.51331778593733928444594361840, −4.47513802839947470737032323861, −2.53623353923091238315535545203, −0.82624711791797162386794860521, 0.82624711791797162386794860521, 2.53623353923091238315535545203, 4.47513802839947470737032323861, 5.51331778593733928444594361840, 7.18845902309268000133166862156, 7.60149056272632292052777748313, 8.90405576737827996770584228794, 10.32794976444752957642728447190, 11.1348867056033258260813644738, 11.77103220035168417289622195039, 12.793928035823971873550369159217, 14.78558369990380964076016332804, 15.658543266406915149414498187174, 16.41218087657527046400338574975, 17.551313996657898695121121330989, 18.11625822723174927712180828683, 19.101041912396701593884952390017, 20.161410190655384488370337217570, 21.11312455363948113257821184828, 22.20091746193614388946818149208, 23.51964211783655315769254609814, 24.10096993852439686588833832324, 24.8797145776674866369859770361, 26.69179915172111168984860069084, 26.91015549878388112275243373098

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