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 + 5·7-s − 8-s + 9-s + 3·10-s + 6·11-s + 12-s − 4·13-s − 5·14-s − 3·15-s + 16-s + 3·17-s − 18-s − 19-s − 3·20-s + 5·21-s − 6·22-s − 24-s + 4·25-s + 4·26-s + 27-s + 5·28-s − 29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.80·11-s + 0.288·12-s − 1.10·13-s − 1.33·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s + 1.09·21-s − 1.27·22-s − 0.204·24-s + 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.944·28-s − 0.185·29-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  =  $(2,\ 174,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.018019957$
$L(\frac12)$  $\approx$  $1.018019957$
$L(\frac{3}{2})$   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 \( 1 - 5 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−19.83332466619983, −19.13701926166968, −18.24919437285145, −17.18721004489085, −16.65666433028828, −15.18961833244927, −14.80532589161956, −14.12215594148932, −12.13132664377911, −11.78922072236665, −10.88227026094639, −9.487370424405801, −8.443424828695416, −7.832681463535283, −6.972311926985583, −4.855122832542796, −3.704355415719718, −1.632723579360293, 1.632723579360293, 3.704355415719718, 4.855122832542796, 6.972311926985583, 7.832681463535283, 8.443424828695416, 9.487370424405801, 10.88227026094639, 11.78922072236665, 12.13132664377911, 14.12215594148932, 14.80532589161956, 15.18961833244927, 16.65666433028828, 17.18721004489085, 18.24919437285145, 19.13701926166968, 19.83332466619983

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