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Properties

Label 2-32340-1.1-c1-0-39
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 − 3·17-s − 19-s − 23-s + 25-s + 27-s + 5·29-s − 33-s − 7·37-s + 6·39-s + 10·41-s − 13·43-s − 45-s + 7·47-s − 3·51-s − 10·53-s + 55-s − 57-s − 13·59-s + 8·61-s − 6·65-s − 14·67-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 − 0.727·17-s − 0.229·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.928·29-s − 0.174·33-s − 1.15·37-s + 0.960·39-s + 1.56·41-s − 1.98·43-s − 0.149·45-s + 1.02·47-s − 0.420·51-s − 1.37·53-s + 0.134·55-s − 0.132·57-s − 1.69·59-s + 1.02·61-s − 0.744·65-s − 1.71·67-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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−15.39531220067708, −14.89783691528350, −14.10751187489564, −13.70494260250322, −13.39517109764220, −12.62600102948220, −12.29604452244079, −11.50177956268922, −11.00139155561796, −10.59545667196588, −10.00427178015648, −9.176653618929539, −8.803663218098920, −8.277817928381827, −7.853730018084263, −7.132892775679953, −6.454795016315896, −6.092202356458464, −5.174688456129949, −4.511940428484288, −3.931418777008522, −3.338737105048007, −2.722832751036097, −1.827596182892884, −1.130346730954779, 0, 1.130346730954779, 1.827596182892884, 2.722832751036097, 3.338737105048007, 3.931418777008522, 4.511940428484288, 5.174688456129949, 6.092202356458464, 6.454795016315896, 7.132892775679953, 7.853730018084263, 8.277817928381827, 8.803663218098920, 9.176653618929539, 10.00427178015648, 10.59545667196588, 11.00139155561796, 11.50177956268922, 12.29604452244079, 12.62600102948220, 13.39517109764220, 13.70494260250322, 14.10751187489564, 14.89783691528350, 15.39531220067708

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