Properties

Degree $2$
Conductor $197$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 3·7-s − 3·9-s + 4·11-s − 2·13-s + 6·14-s − 4·16-s − 8·17-s + 6·18-s − 3·19-s − 8·22-s − 3·23-s − 5·25-s + 4·26-s − 6·28-s + 7·29-s − 10·31-s + 8·32-s + 16·34-s − 6·36-s + 7·37-s + 6·38-s + 9·41-s + 43-s + 8·44-s + 6·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.13·7-s − 9-s + 1.20·11-s − 0.554·13-s + 1.60·14-s − 16-s − 1.94·17-s + 1.41·18-s − 0.688·19-s − 1.70·22-s − 0.625·23-s − 25-s + 0.784·26-s − 1.13·28-s + 1.29·29-s − 1.79·31-s + 1.41·32-s + 2.74·34-s − 36-s + 1.15·37-s + 0.973·38-s + 1.40·41-s + 0.152·43-s + 1.20·44-s + 0.884·46-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(197\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{197} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 197,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.67176147802056, −19.45217455433576, −18.00459917164315, −17.57607192742446, −16.66865302531532, −16.12735739141128, −14.96836717935500, −13.89861678925170, −12.86890346873443, −11.63768790626972, −10.87939044119723, −9.670505384884660, −9.141663261808506, −8.282753935470816, −6.924720413403952, −6.175919408442796, −4.157855191930721, −2.332771160673460, 0, 2.332771160673460, 4.157855191930721, 6.175919408442796, 6.924720413403952, 8.282753935470816, 9.141663261808506, 9.670505384884660, 10.87939044119723, 11.63768790626972, 12.86890346873443, 13.89861678925170, 14.96836717935500, 16.12735739141128, 16.66865302531532, 17.57607192742446, 18.00459917164315, 19.45217455433576, 19.67176147802056

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