Properties

Label 2-32340-1.1-c1-0-35
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

−15.15524902206156, −14.70710193498211, −14.39307096926770, −13.69568561795317, −13.25870292329739, −12.70222359887396, −12.31561175233330, −11.59243923530749, −11.00298190188004, −10.43717824093291, −9.799573191270867, −9.412821509536265, −8.997042099780409, −8.212796587834587, −7.683452793615313, −7.039734083173407, −6.801149635997091, −5.686956508408754, −5.329568750643939, −4.649964108736493, −4.011002632032674, −3.168760627025323, −2.504634139135713, −2.107542343755663, −1.099515140980823, 0, 1.099515140980823, 2.107542343755663, 2.504634139135713, 3.168760627025323, 4.011002632032674, 4.649964108736493, 5.329568750643939, 5.686956508408754, 6.801149635997091, 7.039734083173407, 7.683452793615313, 8.212796587834587, 8.997042099780409, 9.412821509536265, 9.799573191270867, 10.43717824093291, 11.00298190188004, 11.59243923530749, 12.31561175233330, 12.70222359887396, 13.25870292329739, 13.69568561795317, 14.39307096926770, 14.70710193498211, 15.15524902206156

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