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Properties

Label 2-39600-1.1-c1-0-102
Degree $2$
Conductor $39600$
Sign $-1$
Analytic cond. $316.207$
Root an. cond. $17.7822$
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  − 2·7-s + 11-s + 13-s + 8·17-s + 5·19-s − 9·23-s − 5·29-s − 7·31-s + 2·37-s − 2·41-s + 9·43-s + 12·47-s − 3·49-s + 4·53-s − 10·59-s + 2·61-s − 2·67-s − 13·71-s − 14·73-s − 2·77-s + 83-s + 15·89-s − 2·91-s + 7·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.301·11-s + 0.277·13-s + 1.94·17-s + 1.14·19-s − 1.87·23-s − 0.928·29-s − 1.25·31-s + 0.328·37-s − 0.312·41-s + 1.37·43-s + 1.75·47-s − 3/7·49-s + 0.549·53-s − 1.30·59-s + 0.256·61-s − 0.244·67-s − 1.54·71-s − 1.63·73-s − 0.227·77-s + 0.109·83-s + 1.58·89-s − 0.209·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(316.207\)
Root analytic conductor: \(17.7822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 39600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.96382504328522, −14.39052429904731, −14.16801868500856, −13.47500940621815, −13.02891254414275, −12.29175973954378, −12.02660315936278, −11.60159528201868, −10.66119159048812, −10.38276565393576, −9.621874547631977, −9.419321593505048, −8.794990790253545, −7.900826458475517, −7.558783165064087, −7.148554908113161, −6.143981393803648, −5.778969458289672, −5.469572176620477, −4.416793771582130, −3.738244233121806, −3.397317162730485, −2.644207775227720, −1.715095092569139, −1.018252366116494, 0, 1.018252366116494, 1.715095092569139, 2.644207775227720, 3.397317162730485, 3.738244233121806, 4.416793771582130, 5.469572176620477, 5.778969458289672, 6.143981393803648, 7.148554908113161, 7.558783165064087, 7.900826458475517, 8.794990790253545, 9.419321593505048, 9.621874547631977, 10.38276565393576, 10.66119159048812, 11.60159528201868, 12.02660315936278, 12.29175973954378, 13.02891254414275, 13.47500940621815, 14.16801868500856, 14.39052429904731, 14.96382504328522

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