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Properties

Label 2-32340-1.1-c1-0-40
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 − 2·13-s + 15-s − 8·17-s + 2·19-s + 8·23-s + 25-s + 27-s − 33-s + 2·37-s − 2·39-s + 6·43-s + 45-s − 8·47-s − 8·51-s + 6·53-s − 55-s + 2·57-s + 4·59-s − 10·61-s − 2·65-s − 12·67-s + 8·69-s + 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.258·15-s − 1.94·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.914·43-s + 0.149·45-s − 1.16·47-s − 1.12·51-s + 0.824·53-s − 0.134·55-s + 0.264·57-s + 0.520·59-s − 1.28·61-s − 0.248·65-s − 1.46·67-s + 0.963·69-s + 0.949·71-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.31303925137457, −14.74444467222583, −14.33363625443634, −13.55803607977982, −13.31451550591124, −12.83890925598031, −12.30187445622944, −11.29422257760851, −11.25965985278436, −10.37642697487564, −9.981213766298598, −9.203462607168314, −8.929449063942737, −8.455900188070710, −7.490153701708526, −7.247735479219475, −6.556428620678508, −5.966766575246469, −5.115852622552328, −4.675807857349451, −4.069190435811190, −3.066018364577704, −2.669969073633310, −1.981740581218429, −1.143124712480007, 0, 1.143124712480007, 1.981740581218429, 2.669969073633310, 3.066018364577704, 4.069190435811190, 4.675807857349451, 5.115852622552328, 5.966766575246469, 6.556428620678508, 7.247735479219475, 7.490153701708526, 8.455900188070710, 8.929449063942737, 9.203462607168314, 9.981213766298598, 10.37642697487564, 11.25965985278436, 11.29422257760851, 12.30187445622944, 12.83890925598031, 13.31451550591124, 13.55803607977982, 14.33363625443634, 14.74444467222583, 15.31303925137457

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