Properties

Label 2-32340-1.1-c1-0-27
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 + 6·13-s + 15-s + 2·17-s − 2·19-s + 4·23-s + 25-s − 27-s − 6·29-s + 33-s − 2·37-s − 6·39-s − 4·41-s + 4·43-s − 45-s − 2·51-s + 4·53-s + 55-s + 2·57-s − 6·59-s + 2·61-s − 6·65-s + 14·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 0.258·15-s + 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.328·37-s − 0.960·39-s − 0.624·41-s + 0.609·43-s − 0.149·45-s − 0.280·51-s + 0.549·53-s + 0.134·55-s + 0.264·57-s − 0.781·59-s + 0.256·61-s − 0.744·65-s + 1.71·67-s − 0.481·69-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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.42403048708944, −14.87330115265681, −14.30179438562677, −13.57277072792905, −13.14529827890915, −12.73330899968775, −12.08037558486932, −11.50257449022604, −11.06525615427727, −10.66364803260315, −10.12274473244587, −9.332852347843729, −8.788286051955066, −8.292963078328418, −7.645677621180102, −7.065334943110508, −6.472210053722106, −5.839503224193533, −5.387849287913223, −4.663132575852891, −3.905110500030437, −3.517740925470902, −2.671970592263774, −1.639666276901872, −0.9981093122521042, 0, 0.9981093122521042, 1.639666276901872, 2.671970592263774, 3.517740925470902, 3.905110500030437, 4.663132575852891, 5.387849287913223, 5.839503224193533, 6.472210053722106, 7.065334943110508, 7.645677621180102, 8.292963078328418, 8.788286051955066, 9.332852347843729, 10.12274473244587, 10.66364803260315, 11.06525615427727, 11.50257449022604, 12.08037558486932, 12.73330899968775, 13.14529827890915, 13.57277072792905, 14.30179438562677, 14.87330115265681, 15.42403048708944

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