Properties

Label 2-3179-1.1-c1-0-149
Degree $2$
Conductor $3179$
Sign $-1$
Analytic cond. $25.3844$
Root an. cond. $5.03829$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 2·7-s − 2·9-s + 2·10-s − 11-s + 2·12-s + 4·13-s − 4·14-s − 15-s − 4·16-s + 4·18-s − 2·20-s + 2·21-s + 2·22-s + 23-s − 4·25-s − 8·26-s − 5·27-s + 4·28-s + 2·30-s − 7·31-s + 8·32-s − 33-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s + 1.10·13-s − 1.06·14-s − 0.258·15-s − 16-s + 0.942·18-s − 0.447·20-s + 0.436·21-s + 0.426·22-s + 0.208·23-s − 4/5·25-s − 1.56·26-s − 0.962·27-s + 0.755·28-s + 0.365·30-s − 1.25·31-s + 1.41·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3179\)    =    \(11 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(25.3844\)
Root analytic conductor: \(5.03829\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3179,\ (\ :1/2),\ -1)\)

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)$
bad11 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.263155997010663128403798948932, −7.907366117668739819170581630667, −7.29774796427450513185159809590, −6.22323910317192896404427293386, −5.32604921703423297753132504909, −4.24329958875835728994074784592, −3.36204123451452550822417446571, −2.23424739606738503289626266310, −1.35872004735313563095158147020, 0, 1.35872004735313563095158147020, 2.23424739606738503289626266310, 3.36204123451452550822417446571, 4.24329958875835728994074784592, 5.32604921703423297753132504909, 6.22323910317192896404427293386, 7.29774796427450513185159809590, 7.907366117668739819170581630667, 8.263155997010663128403798948932

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