Properties

Label 2-16744-1.1-c1-0-0
Degree $2$
Conductor $16744$
Sign $1$
Analytic cond. $133.701$
Root an. cond. $11.5629$
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  − 2·5-s − 7-s − 3·9-s − 13-s + 2·17-s + 4·19-s − 23-s − 25-s − 2·29-s + 8·31-s + 2·35-s + 2·37-s + 10·41-s − 8·43-s + 6·45-s + 49-s − 6·53-s − 10·61-s + 3·63-s + 2·65-s − 8·67-s + 2·73-s − 16·79-s + 9·81-s − 12·83-s − 4·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 9-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.338·35-s + 0.328·37-s + 1.56·41-s − 1.21·43-s + 0.894·45-s + 1/7·49-s − 0.824·53-s − 1.28·61-s + 0.377·63-s + 0.248·65-s − 0.977·67-s + 0.234·73-s − 1.80·79-s + 81-s − 1.31·83-s − 0.433·85-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16744\)    =    \(2^{3} \cdot 7 \cdot 13 \cdot 23\)
Sign: $1$
Analytic conductor: \(133.701\)
Root analytic conductor: \(11.5629\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{16744} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 16744,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9029884235\)
\(L(\frac12)\) \(\approx\) \(0.9029884235\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
13 \( 1 + T \)
23 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.94571987358128, −15.36152877430790, −14.88010154525163, −14.15916549211672, −13.87908063254225, −13.09675251015080, −12.48406626773659, −11.76663518614509, −11.69145904734800, −10.98347473259634, −10.24963864966512, −9.654698706595248, −9.115267523464065, −8.355190266174915, −7.858767718797310, −7.432376048037043, −6.594093630242149, −5.932487582565742, −5.390339161730453, −4.542929462699503, −3.936386924230372, −3.034344852900483, −2.794453117196429, −1.483219050363493, −0.4118542532625354, 0.4118542532625354, 1.483219050363493, 2.794453117196429, 3.034344852900483, 3.936386924230372, 4.542929462699503, 5.390339161730453, 5.932487582565742, 6.594093630242149, 7.432376048037043, 7.858767718797310, 8.355190266174915, 9.115267523464065, 9.654698706595248, 10.24963864966512, 10.98347473259634, 11.69145904734800, 11.76663518614509, 12.48406626773659, 13.09675251015080, 13.87908063254225, 14.15916549211672, 14.88010154525163, 15.36152877430790, 15.94571987358128

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