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Properties

Label 2-32340-1.1-c1-0-33
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 + 2·13-s − 15-s + 2·17-s − 4·19-s − 6·23-s + 25-s + 27-s − 10·29-s + 2·31-s + 33-s + 2·37-s + 2·39-s − 8·41-s − 45-s + 12·47-s + 2·51-s + 2·53-s − 55-s − 4·57-s + 10·59-s + 4·61-s − 2·65-s − 10·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.359·31-s + 0.174·33-s + 0.328·37-s + 0.320·39-s − 1.24·41-s − 0.149·45-s + 1.75·47-s + 0.280·51-s + 0.274·53-s − 0.134·55-s − 0.529·57-s + 1.30·59-s + 0.512·61-s − 0.248·65-s − 1.22·67-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: Trivial
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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−15.21766041237500, −14.82879887876780, −14.33224581593629, −13.62138663449984, −13.40428972807802, −12.57786046244339, −12.25756471277179, −11.58441915856003, −11.08878539743007, −10.46704144049589, −9.940270185424139, −9.339205950738214, −8.756159982530416, −8.262221872797319, −7.793473562891170, −7.167720539415120, −6.573430613880278, −5.892631201774294, −5.335896791798256, −4.427649533054557, −3.764709653077744, −3.632176190091413, −2.515154896207592, −1.966550496283889, −1.080789185073166, 0, 1.080789185073166, 1.966550496283889, 2.515154896207592, 3.632176190091413, 3.764709653077744, 4.427649533054557, 5.335896791798256, 5.892631201774294, 6.573430613880278, 7.167720539415120, 7.793473562891170, 8.262221872797319, 8.756159982530416, 9.339205950738214, 9.940270185424139, 10.46704144049589, 11.08878539743007, 11.58441915856003, 12.25756471277179, 12.57786046244339, 13.40428972807802, 13.62138663449984, 14.33224581593629, 14.82879887876780, 15.21766041237500

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