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Properties

Label 2-32340-1.1-c1-0-21
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 − 5·13-s − 15-s + 2·17-s − 6·19-s + 2·23-s + 25-s − 27-s − 4·29-s + 9·31-s + 33-s + 6·37-s + 5·39-s − 12·41-s + 43-s + 45-s + 2·47-s − 2·51-s + 6·53-s − 55-s + 6·57-s + 59-s + 6·61-s − 5·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.258·15-s + 0.485·17-s − 1.37·19-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.61·31-s + 0.174·33-s + 0.986·37-s + 0.800·39-s − 1.87·41-s + 0.152·43-s + 0.149·45-s + 0.291·47-s − 0.280·51-s + 0.824·53-s − 0.134·55-s + 0.794·57-s + 0.130·59-s + 0.768·61-s − 0.620·65-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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−15.33622282003923, −14.74902679143957, −14.39791892102235, −13.62969199021306, −13.08168871277998, −12.74629200197067, −12.05131699272000, −11.72829988598300, −11.01881106233913, −10.43054982460551, −9.957152940775583, −9.669426976945611, −8.786860925201238, −8.289996308207007, −7.607650767423557, −6.972950938038469, −6.549191229568247, −5.814223113621227, −5.324618891500363, −4.680573835207944, −4.230410329567030, −3.233718070860852, −2.486888893480561, −1.942881116616884, −0.9097553963726811, 0, 0.9097553963726811, 1.942881116616884, 2.486888893480561, 3.233718070860852, 4.230410329567030, 4.680573835207944, 5.324618891500363, 5.814223113621227, 6.549191229568247, 6.972950938038469, 7.607650767423557, 8.289996308207007, 8.786860925201238, 9.669426976945611, 9.957152940775583, 10.43054982460551, 11.01881106233913, 11.72829988598300, 12.05131699272000, 12.74629200197067, 13.08168871277998, 13.62969199021306, 14.39791892102235, 14.74902679143957, 15.33622282003923

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