Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 3·5-s + 6-s − 3·7-s − 8-s + 9-s − 3·10-s + 6·11-s − 12-s + 3·14-s − 3·15-s + 16-s + 7·17-s − 18-s + 5·19-s + 3·20-s + 3·21-s − 6·22-s − 8·23-s + 24-s + 4·25-s − 27-s − 3·28-s + 29-s + 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.80·11-s − 0.288·12-s + 0.801·14-s − 0.774·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.14·19-s + 0.670·20-s + 0.654·21-s − 1.27·22-s − 1.66·23-s + 0.204·24-s + 4/5·25-s − 0.192·27-s − 0.566·28-s + 0.185·29-s + 0.547·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(174\)    =    \(2 \cdot 3 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{174} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 174,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8687758076$
$L(\frac12)$  $\approx$  $0.8687758076$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.77961569736558, −18.81794002729217, −18.08137801092174, −17.16830535667390, −16.66752952222184, −15.94955283275213, −14.42420535171882, −13.78453454056522, −12.41495073006028, −11.84547600245529, −10.35936278010760, −9.685450630995214, −9.168917497360859, −7.381503836704062, −6.258829091683767, −5.716580077602401, −3.506789222233516, −1.495674029995115, 1.495674029995115, 3.506789222233516, 5.716580077602401, 6.258829091683767, 7.381503836704062, 9.168917497360859, 9.685450630995214, 10.35936278010760, 11.84547600245529, 12.41495073006028, 13.78453454056522, 14.42420535171882, 15.94955283275213, 16.66752952222184, 17.16830535667390, 18.08137801092174, 18.81794002729217, 19.77961569736558

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