Properties

Label 2-158400-1.1-c1-0-79
Degree $2$
Conductor $158400$
Sign $1$
Analytic cond. $1264.83$
Root an. cond. $35.5644$
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  − 7-s − 11-s − 4·13-s − 3·17-s + 5·19-s + 23-s − 2·29-s + 2·31-s + 5·37-s − 9·41-s + 12·43-s + 11·47-s − 6·49-s + 59-s + 5·71-s − 16·73-s + 77-s + 11·79-s − 6·83-s + 4·89-s + 4·91-s + 15·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.301·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s + 0.208·23-s − 0.371·29-s + 0.359·31-s + 0.821·37-s − 1.40·41-s + 1.82·43-s + 1.60·47-s − 6/7·49-s + 0.130·59-s + 0.593·71-s − 1.87·73-s + 0.113·77-s + 1.23·79-s − 0.658·83-s + 0.423·89-s + 0.419·91-s + 1.52·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(158400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1264.83\)
Root analytic conductor: \(35.5644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{158400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 158400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.704795268\)
\(L(\frac12)\) \(\approx\) \(1.704795268\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−13.33738439237458, −12.76470570609380, −12.38066431986992, −11.88918624405390, −11.41057000248344, −10.96375576109054, −10.29247419240479, −10.01294104534609, −9.392960358149144, −9.101201101844788, −8.529954142471679, −7.772337565652639, −7.505696237878900, −7.005226267596854, −6.483926394725155, −5.783856068148548, −5.466464889580414, −4.671896262899467, −4.477918908269462, −3.604139698335403, −3.099754906952647, −2.484146860401044, −2.046200874745059, −1.083432575126823, −0.4168921890604990, 0.4168921890604990, 1.083432575126823, 2.046200874745059, 2.484146860401044, 3.099754906952647, 3.604139698335403, 4.477918908269462, 4.671896262899467, 5.466464889580414, 5.783856068148548, 6.483926394725155, 7.005226267596854, 7.505696237878900, 7.772337565652639, 8.529954142471679, 9.101201101844788, 9.392960358149144, 10.01294104534609, 10.29247419240479, 10.96375576109054, 11.41057000248344, 11.88918624405390, 12.38066431986992, 12.76470570609380, 13.33738439237458

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