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Properties

Label 2-109200-1.1-c1-0-166
Degree $2$
Conductor $109200$
Sign $-1$
Analytic cond. $871.966$
Root an. cond. $29.5290$
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 − 7-s + 9-s + 4·11-s − 13-s + 6·17-s − 2·19-s − 21-s + 2·23-s + 27-s + 2·29-s − 6·31-s + 4·33-s − 2·37-s − 39-s − 4·41-s − 6·43-s + 49-s + 6·51-s − 4·53-s − 2·57-s − 6·59-s + 2·61-s − 63-s − 8·67-s + 2·69-s + 16·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 1.45·17-s − 0.458·19-s − 0.218·21-s + 0.417·23-s + 0.192·27-s + 0.371·29-s − 1.07·31-s + 0.696·33-s − 0.328·37-s − 0.160·39-s − 0.624·41-s − 0.914·43-s + 1/7·49-s + 0.840·51-s − 0.549·53-s − 0.264·57-s − 0.781·59-s + 0.256·61-s − 0.125·63-s − 0.977·67-s + 0.240·69-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(109200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(871.966\)
Root analytic conductor: \(29.5290\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 109200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
13 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.79642836471294, −13.66123539997956, −12.83332217340194, −12.41684516599547, −12.16555585260863, −11.47885568113306, −11.03577773960039, −10.29304513568465, −9.979500555342588, −9.423427150338253, −9.042886279761778, −8.502346116844575, −8.009361699430483, −7.343236208040126, −7.021646159877197, −6.346033651663642, −5.965112975938372, −5.146750976728492, −4.743318048550891, −3.899699943551269, −3.512433468776214, −3.091847565862664, −2.268738523580166, −1.577971579074268, −1.043427853757867, 0, 1.043427853757867, 1.577971579074268, 2.268738523580166, 3.091847565862664, 3.512433468776214, 3.899699943551269, 4.743318048550891, 5.146750976728492, 5.965112975938372, 6.346033651663642, 7.021646159877197, 7.343236208040126, 8.009361699430483, 8.502346116844575, 9.042886279761778, 9.423427150338253, 9.979500555342588, 10.29304513568465, 11.03577773960039, 11.47885568113306, 12.16555585260863, 12.41684516599547, 12.83332217340194, 13.66123539997956, 13.79642836471294

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