Properties

Label 2-120e2-1.1-c1-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·13-s + 8·17-s − 4·29-s − 2·37-s + 8·41-s − 7·49-s + 4·53-s + 10·61-s − 6·73-s − 16·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.66·13-s + 1.94·17-s − 0.742·29-s − 0.328·37-s + 1.24·41-s − 49-s + 0.549·53-s + 1.28·61-s − 0.702·73-s − 1.69·89-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: \(0\)
Selberg data: \((2,\ 14400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.781996718\)
\(L(\frac12)\) \(\approx\) \(1.781996718\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−16.26270648035631, −15.50999581467826, −14.86920729152731, −14.35226644160998, −14.22286082238002, −13.16215763577493, −12.73078851081897, −12.13073873704581, −11.77006813113095, −11.00964528298145, −10.29874397864088, −9.740958698420582, −9.485134824932497, −8.546499042769316, −7.866159085841111, −7.403620027755322, −6.896370513678137, −5.887131394337717, −5.431390921798599, −4.797920452177807, −3.991657332270240, −3.203305498468879, −2.535629709683144, −1.636959756070308, −0.5879881805080925, 0.5879881805080925, 1.636959756070308, 2.535629709683144, 3.203305498468879, 3.991657332270240, 4.797920452177807, 5.431390921798599, 5.887131394337717, 6.896370513678137, 7.403620027755322, 7.866159085841111, 8.546499042769316, 9.485134824932497, 9.740958698420582, 10.29874397864088, 11.00964528298145, 11.77006813113095, 12.13073873704581, 12.73078851081897, 13.16215763577493, 14.22286082238002, 14.35226644160998, 14.86920729152731, 15.50999581467826, 16.26270648035631

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