Properties

Label 2-159120-1.1-c1-0-105
Degree $2$
Conductor $159120$
Sign $-1$
Analytic cond. $1270.57$
Root an. cond. $35.6451$
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  − 5-s + 4·11-s + 13-s − 17-s + 4·19-s + 25-s + 2·29-s + 6·37-s + 6·41-s + 4·43-s − 7·49-s − 6·53-s − 4·55-s + 4·59-s + 6·61-s − 65-s − 12·67-s − 16·71-s − 6·73-s − 8·79-s + 12·83-s + 85-s − 2·89-s − 4·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.768·61-s − 0.124·65-s − 1.46·67-s − 1.89·71-s − 0.702·73-s − 0.900·79-s + 1.31·83-s + 0.108·85-s − 0.211·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(1270.57\)
Root analytic conductor: \(35.6451\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{159120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 159120,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.42187975240044, −13.12969223944762, −12.49821228838734, −11.98809846898737, −11.69303853307512, −11.15702775481924, −10.86726407227865, −10.05782583447022, −9.722810902987563, −9.111168459854545, −8.823830948624291, −8.221651067782115, −7.613165114490871, −7.292518391794855, −6.677648505248490, −6.111477401528983, −5.813047858159107, −4.979479584416574, −4.428926329665892, −4.082994119861965, −3.375731189071332, −2.956516477856301, −2.201211178055238, −1.337519297013855, −0.9841385931023228, 0, 0.9841385931023228, 1.337519297013855, 2.201211178055238, 2.956516477856301, 3.375731189071332, 4.082994119861965, 4.428926329665892, 4.979479584416574, 5.813047858159107, 6.111477401528983, 6.677648505248490, 7.292518391794855, 7.613165114490871, 8.221651067782115, 8.823830948624291, 9.111168459854545, 9.722810902987563, 10.05782583447022, 10.86726407227865, 11.15702775481924, 11.69303853307512, 11.98809846898737, 12.49821228838734, 13.12969223944762, 13.42187975240044

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