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 + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 6·11-s − 12-s − 4·13-s + 14-s − 15-s + 16-s − 7·17-s + 18-s − 3·19-s + 20-s − 21-s + 6·22-s + 4·23-s − 24-s − 4·25-s − 4·26-s − 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 0.688·19-s + 0.223·20-s − 0.218·21-s + 1.27·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.784·26-s − 0.192·27-s + 0.188·28-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  =  $(2,\ 174,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.584700771$
$L(\frac12)$  $\approx$  $1.584700771$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−19.76336990398227, −19.32546805969097, −17.71716440258373, −17.32097787492229, −16.51645105982915, −15.24339405072183, −14.61226782457019, −13.65380110349327, −12.69382741216715, −11.73639879382114, −11.10440562935940, −9.836098981125503, −8.771228911258239, −7.035923832171918, −6.379372565841810, −5.047778846759627, −4.050202157152346, −1.991938748589375, 1.991938748589375, 4.050202157152346, 5.047778846759627, 6.379372565841810, 7.035923832171918, 8.771228911258239, 9.836098981125503, 11.10440562935940, 11.73639879382114, 12.69382741216715, 13.65380110349327, 14.61226782457019, 15.24339405072183, 16.51645105982915, 17.32097787492229, 17.71716440258373, 19.32546805969097, 19.76336990398227

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