Properties

Label 2-344760-1.1-c1-0-45
Degree $2$
Conductor $344760$
Sign $-1$
Analytic cond. $2752.92$
Root an. cond. $52.4682$
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 + 5-s + 2·7-s + 9-s − 4·11-s − 15-s + 17-s + 4·19-s − 2·21-s + 25-s − 27-s + 8·31-s + 4·33-s + 2·35-s + 2·37-s − 2·41-s − 6·43-s + 45-s − 12·47-s − 3·49-s − 51-s + 12·53-s − 4·55-s − 4·57-s + 2·63-s + 12·67-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.43·31-s + 0.696·33-s + 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.914·43-s + 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.140·51-s + 1.64·53-s − 0.539·55-s − 0.529·57-s + 0.251·63-s + 1.46·67-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(344760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2752.92\)
Root analytic conductor: \(52.4682\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 344760,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.94568744632769, −12.17078094063677, −11.78053514320365, −11.56800758185494, −10.94942956832072, −10.52353611722286, −10.03831704528576, −9.843055543360987, −9.245292913382313, −8.467791378326921, −8.247968568731865, −7.751482387467634, −7.259077756224003, −6.745663272493530, −6.222123264187902, −5.662069607994103, −5.307958204954907, −4.741563612096554, −4.634890655332637, −3.673586846695884, −3.141478327237891, −2.577590639405045, −1.991981345572964, −1.361512506569495, −0.8136921604492044, 0, 0.8136921604492044, 1.361512506569495, 1.991981345572964, 2.577590639405045, 3.141478327237891, 3.673586846695884, 4.634890655332637, 4.741563612096554, 5.307958204954907, 5.662069607994103, 6.222123264187902, 6.745663272493530, 7.259077756224003, 7.751482387467634, 8.247968568731865, 8.467791378326921, 9.245292913382313, 9.843055543360987, 10.03831704528576, 10.52353611722286, 10.94942956832072, 11.56800758185494, 11.78053514320365, 12.17078094063677, 12.94568744632769

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