Properties

Label 2-11424-1.1-c1-0-20
Degree $2$
Conductor $11424$
Sign $-1$
Analytic cond. $91.2210$
Root an. cond. $9.55097$
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  + 3-s + 2·5-s − 7-s + 9-s − 2·13-s + 2·15-s − 17-s − 8·19-s − 21-s − 25-s + 27-s + 2·29-s − 2·35-s + 2·37-s − 2·39-s + 6·41-s + 4·43-s + 2·45-s − 8·47-s + 49-s − 51-s − 6·53-s − 8·57-s + 4·59-s + 6·61-s − 63-s − 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.242·17-s − 1.83·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.338·35-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s − 1.05·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s − 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11424\)    =    \(2^{5} \cdot 3 \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(91.2210\)
Root analytic conductor: \(9.55097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11424} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11424,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.87079375402046, −16.05813291483134, −15.62638421325325, −14.83247913066691, −14.50664808886006, −13.92299163733816, −13.28875102018074, −12.75636271978870, −12.49080676858359, −11.46457459727792, −10.86101518545347, −10.20325947359629, −9.660806365771508, −9.280352065884201, −8.438994334244852, −8.064162134913061, −7.087759127546005, −6.559189554861878, −5.984679268880189, −5.208838067092040, −4.370326621113784, −3.800309786716256, −2.649020663617757, −2.337037109474863, −1.400885418571368, 0, 1.400885418571368, 2.337037109474863, 2.649020663617757, 3.800309786716256, 4.370326621113784, 5.208838067092040, 5.984679268880189, 6.559189554861878, 7.087759127546005, 8.064162134913061, 8.438994334244852, 9.280352065884201, 9.660806365771508, 10.20325947359629, 10.86101518545347, 11.46457459727792, 12.49080676858359, 12.75636271978870, 13.28875102018074, 13.92299163733816, 14.50664808886006, 14.83247913066691, 15.62638421325325, 16.05813291483134, 16.87079375402046

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