Properties

Label 2-344760-1.1-c1-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 5·7-s + 9-s − 3·11-s − 15-s − 17-s + 7·19-s − 5·21-s − 6·23-s + 25-s − 27-s + 9·29-s − 6·31-s + 3·33-s + 5·35-s + 9·37-s − 3·41-s − 6·43-s + 45-s − 9·47-s + 18·49-s + 51-s + 3·53-s − 3·55-s − 7·57-s + 14·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.904·11-s − 0.258·15-s − 0.242·17-s + 1.60·19-s − 1.09·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s − 1.07·31-s + 0.522·33-s + 0.845·35-s + 1.47·37-s − 0.468·41-s − 0.914·43-s + 0.149·45-s − 1.31·47-s + 18/7·49-s + 0.140·51-s + 0.412·53-s − 0.404·55-s − 0.927·57-s + 1.82·59-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: \(0\)
Selberg data: \((2,\ 344760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.600316118\)
\(L(\frac12)\) \(\approx\) \(3.600316118\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.58319553033756, −11.86830609317857, −11.63096679162888, −11.30336584901465, −10.82410958261957, −10.31536038280204, −9.918336689746741, −9.551997150912615, −8.842448656268757, −8.166687650571060, −8.037665262933213, −7.699215736587294, −6.858773633322453, −6.666700486145507, −5.813350612901152, −5.379541196158304, −5.125835790464960, −4.754349136817391, −4.096850809379740, −3.567554597199031, −2.698481797250125, −2.241484644412868, −1.708546185867027, −1.099466486473567, −0.5599534017638584, 0.5599534017638584, 1.099466486473567, 1.708546185867027, 2.241484644412868, 2.698481797250125, 3.567554597199031, 4.096850809379740, 4.754349136817391, 5.125835790464960, 5.379541196158304, 5.813350612901152, 6.666700486145507, 6.858773633322453, 7.699215736587294, 8.037665262933213, 8.166687650571060, 8.842448656268757, 9.551997150912615, 9.918336689746741, 10.31536038280204, 10.82410958261957, 11.30336584901465, 11.63096679162888, 11.86830609317857, 12.58319553033756

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