Properties

Label 2-38640-1.1-c1-0-51
Degree $2$
Conductor $38640$
Sign $-1$
Analytic cond. $308.541$
Root an. cond. $17.5653$
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 + 7-s + 9-s − 4·11-s + 2·13-s − 15-s − 2·17-s + 8·19-s − 21-s − 23-s + 25-s − 27-s − 2·29-s + 4·31-s + 4·33-s + 35-s + 6·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 12·47-s + 49-s + 2·51-s − 10·53-s − 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 1.83·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(308.541\)
Root analytic conductor: \(17.5653\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38640,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.35631144646795, −14.47567206455078, −13.87259389531438, −13.57847521545687, −13.06630026428798, −12.47954172184856, −11.88364393838454, −11.39964423110283, −10.87575786706488, −10.45812875933501, −9.759474927519005, −9.471581086760369, −8.615769270149184, −8.111453383446645, −7.466087063535518, −7.092465379044973, −6.129754461075570, −5.876403786903979, −5.128324065175885, −4.813452379525842, −4.012617953817313, −3.129245500143724, −2.609399551640239, −1.673991271603112, −1.036955669871008, 0, 1.036955669871008, 1.673991271603112, 2.609399551640239, 3.129245500143724, 4.012617953817313, 4.813452379525842, 5.128324065175885, 5.876403786903979, 6.129754461075570, 7.092465379044973, 7.466087063535518, 8.111453383446645, 8.615769270149184, 9.471581086760369, 9.759474927519005, 10.45812875933501, 10.87575786706488, 11.39964423110283, 11.88364393838454, 12.47954172184856, 13.06630026428798, 13.57847521545687, 13.87259389531438, 14.47567206455078, 15.35631144646795

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