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Properties

Label 2-120e2-1.1-c1-0-121
Degree $2$
Conductor $14400$
Sign $-1$
Analytic cond. $114.984$
Root an. cond. $10.7230$
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  + 4·7-s − 2·13-s − 6·17-s − 4·23-s − 2·29-s − 8·31-s + 6·37-s + 6·41-s + 12·43-s − 12·47-s + 9·49-s + 10·53-s + 8·59-s + 10·61-s − 12·67-s − 8·71-s − 10·73-s + 16·79-s − 12·83-s + 6·89-s − 8·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.554·13-s − 1.45·17-s − 0.834·23-s − 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s + 1.82·43-s − 1.75·47-s + 9/7·49-s + 1.37·53-s + 1.04·59-s + 1.28·61-s − 1.46·67-s − 0.949·71-s − 1.17·73-s + 1.80·79-s − 1.31·83-s + 0.635·89-s − 0.838·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(114.984\)
Root analytic conductor: \(10.7230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−16.25982412412648, −15.97908781187501, −15.03326687423519, −14.65750285490969, −14.44343657562160, −13.52449082514230, −13.14588704310483, −12.42000753737975, −11.77882977699796, −11.18499345098731, −10.97259720138072, −10.18157758586617, −9.444768105655394, −8.855882060879666, −8.326459252284200, −7.581459584195255, −7.283725105830698, −6.357597776312737, −5.629727833823217, −5.052097589078270, −4.282340785909847, −3.960793760079985, −2.579342881550402, −2.126124493961775, −1.266047971735512, 0, 1.266047971735512, 2.126124493961775, 2.579342881550402, 3.960793760079985, 4.282340785909847, 5.052097589078270, 5.629727833823217, 6.357597776312737, 7.283725105830698, 7.581459584195255, 8.326459252284200, 8.855882060879666, 9.444768105655394, 10.18157758586617, 10.97259720138072, 11.18499345098731, 11.77882977699796, 12.42000753737975, 13.14588704310483, 13.52449082514230, 14.44343657562160, 14.65750285490969, 15.03326687423519, 15.97908781187501, 16.25982412412648

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