Properties

Label 2-32340-1.1-c1-0-29
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s + 2·19-s − 9·23-s + 25-s + 27-s − 9·29-s + 8·31-s − 33-s + 2·37-s − 39-s + 3·41-s − 43-s − 45-s + 9·47-s + 9·53-s + 55-s + 2·57-s + 12·59-s − 10·61-s + 65-s − 4·67-s − 9·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.458·19-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s + 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.160·39-s + 0.468·41-s − 0.152·43-s − 0.149·45-s + 1.31·47-s + 1.23·53-s + 0.134·55-s + 0.264·57-s + 1.56·59-s − 1.28·61-s + 0.124·65-s − 0.488·67-s − 1.08·69-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: \(1\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.25704974090104, −14.86176805877055, −14.18922074031476, −13.79921323009791, −13.28134894146807, −12.69404769363178, −12.05155925063664, −11.76008026782331, −11.06442236853070, −10.40469212334312, −9.907030897404376, −9.458485639102814, −8.732160229582959, −8.246704628494050, −7.640467888218816, −7.354952862674407, −6.564973707142028, −5.812003458364982, −5.364963092048549, −4.358689056342134, −4.073004908522602, −3.338479670652125, −2.557516692415811, −2.035415134976877, −1.019251313104037, 0, 1.019251313104037, 2.035415134976877, 2.557516692415811, 3.338479670652125, 4.073004908522602, 4.358689056342134, 5.364963092048549, 5.812003458364982, 6.564973707142028, 7.354952862674407, 7.640467888218816, 8.246704628494050, 8.732160229582959, 9.458485639102814, 9.907030897404376, 10.40469212334312, 11.06442236853070, 11.76008026782331, 12.05155925063664, 12.69404769363178, 13.28134894146807, 13.79921323009791, 14.18922074031476, 14.86176805877055, 15.25704974090104

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