Properties

Label 2-16744-1.1-c1-0-1
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  + 2·3-s − 2·5-s + 7-s + 9-s − 5·11-s − 13-s − 4·15-s − 7·17-s + 2·21-s − 23-s − 25-s − 4·27-s + 5·29-s − 3·31-s − 10·33-s − 2·35-s − 37-s − 2·39-s + 3·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s − 14·51-s + 6·53-s + 10·55-s + 7·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 1.03·15-s − 1.69·17-s + 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.769·27-s + 0.928·29-s − 0.538·31-s − 1.74·33-s − 0.338·35-s − 0.164·37-s − 0.320·39-s + 0.468·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 1.96·51-s + 0.824·53-s + 1.34·55-s + 0.911·59-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 16744,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387414311\)
\(L(\frac12)\) \(\approx\) \(1.387414311\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−15.81378931927327, −15.16745276508797, −15.09484398498085, −14.14387307692264, −13.81860017915858, −13.15679430312360, −12.78199061810411, −11.98218035156882, −11.38796644981885, −10.90490974081522, −10.25319036708967, −9.642869213557354, −8.711310219679875, −8.568612037233190, −7.963767370787427, −7.400957662525971, −6.957145263778986, −5.900592400214907, −5.196600268717392, −4.419501128398939, −3.969042883106258, −3.074526321336766, −2.478010947011531, −1.958713327725291, −0.4384132467255143, 0.4384132467255143, 1.958713327725291, 2.478010947011531, 3.074526321336766, 3.969042883106258, 4.419501128398939, 5.196600268717392, 5.900592400214907, 6.957145263778986, 7.400957662525971, 7.963767370787427, 8.568612037233190, 8.711310219679875, 9.642869213557354, 10.25319036708967, 10.90490974081522, 11.38796644981885, 11.98218035156882, 12.78199061810411, 13.15679430312360, 13.81860017915858, 14.14387307692264, 15.09484398498085, 15.16745276508797, 15.81378931927327

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