Properties

Label 2-154869-1.1-c1-0-0
Degree $2$
Conductor $154869$
Sign $1$
Analytic cond. $1236.63$
Root an. cond. $35.1658$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 3·8-s + 9-s − 2·10-s − 11-s + 12-s − 13-s + 2·15-s − 16-s − 6·17-s + 18-s + 2·20-s − 22-s − 8·23-s + 3·24-s − 25-s − 26-s − 27-s + 10·29-s + 2·30-s + 5·32-s + 33-s − 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.447·20-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.85·29-s + 0.365·30-s + 0.883·32-s + 0.174·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154869 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154869\)    =    \(3 \cdot 11 \cdot 13 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1236.63\)
Root analytic conductor: \(35.1658\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 154869,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4813246428\)
\(L(\frac12)\) \(\approx\) \(0.4813246428\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.36052712159147, −12.82649027407667, −12.23644947564567, −11.96371036779809, −11.66533410714943, −11.09876688780574, −10.41867850860123, −10.05065370344546, −9.633959698273725, −8.732213702952654, −8.487456008805251, −8.124180651376678, −7.343643312720002, −6.789158812903874, −6.447249969234959, −5.774475785945141, −5.241619965887037, −4.828834166593596, −4.208245763564248, −3.942014944755907, −3.420235509725336, −2.492559384857799, −2.128046306003076, −0.9413765074043985, −0.2280882459598334, 0.2280882459598334, 0.9413765074043985, 2.128046306003076, 2.492559384857799, 3.420235509725336, 3.942014944755907, 4.208245763564248, 4.828834166593596, 5.241619965887037, 5.774475785945141, 6.447249969234959, 6.789158812903874, 7.343643312720002, 8.124180651376678, 8.487456008805251, 8.732213702952654, 9.633959698273725, 10.05065370344546, 10.41867850860123, 11.09876688780574, 11.66533410714943, 11.96371036779809, 12.23644947564567, 12.82649027407667, 13.36052712159147

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