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Properties

Label 2-32340-1.1-c1-0-15
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 − 6·17-s − 2·19-s − 6·23-s + 25-s − 27-s + 7·31-s + 33-s + 2·37-s + 5·39-s − 43-s − 45-s − 6·47-s + 6·51-s + 6·53-s + 55-s + 2·57-s + 3·59-s + 10·61-s + 5·65-s + 2·67-s + 6·69-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 − 1.45·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.25·31-s + 0.174·33-s + 0.328·37-s + 0.800·39-s − 0.152·43-s − 0.149·45-s − 0.875·47-s + 0.840·51-s + 0.824·53-s + 0.134·55-s + 0.264·57-s + 0.390·59-s + 1.28·61-s + 0.620·65-s + 0.244·67-s + 0.722·69-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: 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−15.28298032053090, −14.94674212345939, −14.28054428663801, −13.69866422145749, −13.07686214936237, −12.64852108234764, −12.05476049712955, −11.60867788249227, −11.20414217393622, −10.42362800938876, −10.08317240102792, −9.522115252214097, −8.784627931162626, −8.153607196012472, −7.742454742662336, −6.914422014503335, −6.636824800748228, −5.921561359740773, −5.176955105611110, −4.601755708304746, −4.230758397944188, −3.383995546119385, −2.359367316363612, −2.095207316646782, −0.7296619606818256, 0, 0.7296619606818256, 2.095207316646782, 2.359367316363612, 3.383995546119385, 4.230758397944188, 4.601755708304746, 5.176955105611110, 5.921561359740773, 6.636824800748228, 6.914422014503335, 7.742454742662336, 8.153607196012472, 8.784627931162626, 9.522115252214097, 10.08317240102792, 10.42362800938876, 11.20414217393622, 11.60867788249227, 12.05476049712955, 12.64852108234764, 13.07686214936237, 13.69866422145749, 14.28054428663801, 14.94674212345939, 15.28298032053090

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