Properties

Label 2-32340-1.1-c1-0-19
Degree $2$
Conductor $32340$
Sign $1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
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 + 9-s + 11-s + 15-s + 6·17-s + 6·19-s − 4·23-s + 25-s − 27-s + 8·29-s + 8·31-s − 33-s + 6·37-s + 4·41-s − 45-s + 12·47-s − 6·51-s − 6·53-s − 55-s − 6·57-s + 4·59-s + 10·61-s + 4·67-s + 4·69-s + 8·71-s + 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.258·15-s + 1.45·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s − 0.174·33-s + 0.986·37-s + 0.624·41-s − 0.149·45-s + 1.75·47-s − 0.840·51-s − 0.824·53-s − 0.134·55-s − 0.794·57-s + 0.520·59-s + 1.28·61-s + 0.488·67-s + 0.481·69-s + 0.949·71-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{32340} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.491883181\)
\(L(\frac12)\) \(\approx\) \(2.491883181\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
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

−15.14653668570914, −14.38121748035254, −14.03888747238616, −13.60449145622436, −12.73546785564276, −12.19049265499435, −11.96191194513903, −11.48314422604312, −10.80200198081557, −10.20157837125669, −9.744570002277927, −9.314307387877490, −8.299166663168836, −8.002967726399217, −7.424020238756841, −6.762671470356621, −6.165109334536206, −5.593845770478931, −5.006510001986142, −4.344367578417473, −3.711902932319374, −3.037772784715677, −2.292277455922247, −0.9732889300160512, −0.8637200904895699, 0.8637200904895699, 0.9732889300160512, 2.292277455922247, 3.037772784715677, 3.711902932319374, 4.344367578417473, 5.006510001986142, 5.593845770478931, 6.165109334536206, 6.762671470356621, 7.424020238756841, 8.002967726399217, 8.299166663168836, 9.314307387877490, 9.744570002277927, 10.20157837125669, 10.80200198081557, 11.48314422604312, 11.96191194513903, 12.19049265499435, 12.73546785564276, 13.60449145622436, 14.03888747238616, 14.38121748035254, 15.14653668570914

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