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Properties

Label 2-154800-1.1-c1-0-93
Degree $2$
Conductor $154800$
Sign $-1$
Analytic cond. $1236.08$
Root an. cond. $35.1579$
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  − 4·13-s + 4·17-s − 4·23-s + 2·29-s − 4·31-s + 6·37-s − 2·41-s + 43-s − 7·49-s − 10·53-s − 8·59-s + 8·61-s + 8·67-s + 12·71-s + 10·73-s − 8·79-s + 14·83-s − 12·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.10·13-s + 0.970·17-s − 0.834·23-s + 0.371·29-s − 0.718·31-s + 0.986·37-s − 0.312·41-s + 0.152·43-s − 49-s − 1.37·53-s − 1.04·59-s + 1.02·61-s + 0.977·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1.53·83-s − 1.27·89-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154800\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(1236.08\)
Root analytic conductor: \(35.1579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 154800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
43 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−13.74604167000952, −12.78815656281048, −12.61598214206762, −12.29735388515917, −11.59157413088430, −11.22930393962972, −10.74249599395741, −9.976427147321724, −9.783596157550720, −9.437643945833887, −8.687779972172486, −8.090627209598609, −7.770527474790146, −7.339821533666933, −6.556209203891439, −6.339202468774792, −5.449653422023511, −5.239308172454074, −4.566520391335966, −4.008919013001405, −3.387633961742566, −2.835363713853038, −2.193504315900839, −1.608593966645035, −0.7805282657606914, 0, 0.7805282657606914, 1.608593966645035, 2.193504315900839, 2.835363713853038, 3.387633961742566, 4.008919013001405, 4.566520391335966, 5.239308172454074, 5.449653422023511, 6.339202468774792, 6.556209203891439, 7.339821533666933, 7.770527474790146, 8.090627209598609, 8.687779972172486, 9.437643945833887, 9.783596157550720, 9.976427147321724, 10.74249599395741, 11.22930393962972, 11.59157413088430, 12.29735388515917, 12.61598214206762, 12.78815656281048, 13.74604167000952

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