Properties

Degree $2$
Conductor $279174$
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 + 2·5-s + 6-s − 7-s + 8-s + 9-s + 2·10-s + 4·11-s + 12-s − 2·13-s − 14-s + 2·15-s + 16-s + 18-s + 4·19-s + 2·20-s − 21-s + 4·22-s + 23-s + 24-s − 25-s − 2·26-s + 27-s − 28-s + 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(279174\)    =    \(2 \cdot 3 \cdot 7 \cdot 17^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{279174} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 279174,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.836790491\)
\(L(\frac12)\) \(\approx\) \(8.836790491\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
23 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.90564885854298, −12.29493994976781, −11.89852534251655, −11.60317113896605, −10.98302092347412, −10.22754605098510, −10.03234763313076, −9.438569543710487, −9.329806513767281, −8.517767481140061, −8.160700068911703, −7.521491547686677, −6.876960677727250, −6.629126709378917, −6.261667202582952, −5.370971014784051, −5.322225316473749, −4.555345475890601, −3.914577928392732, −3.648530309219692, −2.792377500321043, −2.631876900463354, −1.833873871686982, −1.353652762695797, −0.6721208835307005, 0.6721208835307005, 1.353652762695797, 1.833873871686982, 2.631876900463354, 2.792377500321043, 3.648530309219692, 3.914577928392732, 4.555345475890601, 5.322225316473749, 5.370971014784051, 6.261667202582952, 6.629126709378917, 6.876960677727250, 7.521491547686677, 8.160700068911703, 8.517767481140061, 9.329806513767281, 9.438569543710487, 10.03234763313076, 10.22754605098510, 10.98302092347412, 11.60317113896605, 11.89852534251655, 12.29493994976781, 12.90564885854298

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