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Properties

Label 2-79560-1.1-c1-0-17
Degree $2$
Conductor $79560$
Sign $-1$
Analytic cond. $635.289$
Root an. cond. $25.2049$
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 + 8·31-s − 10·37-s − 10·41-s + 4·43-s + 8·47-s − 7·49-s + 2·53-s + 4·55-s + 4·59-s + 6·61-s − 65-s − 4·67-s − 14·73-s + 12·83-s + 85-s + 14·89-s + 4·95-s + 10·97-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 + 1.43·31-s − 1.64·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.274·53-s + 0.539·55-s + 0.520·59-s + 0.768·61-s − 0.124·65-s − 0.488·67-s − 1.63·73-s + 1.31·83-s + 0.108·85-s + 1.48·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(79560\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(635.289\)
Root analytic conductor: \(25.2049\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 79560,\ (\ :1/2),\ -1)\)

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 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.18634526286317, −13.65365114636965, −13.31215556135623, −12.75197478838588, −12.29589171824976, −11.73295423372671, −11.35422108575262, −10.53340544278546, −10.38326704298579, −9.981896112707722, −8.974590052293777, −8.662542851706668, −8.245266000386979, −7.606105146395709, −7.199254644593773, −6.416923276727690, −6.167992055291974, −5.155430800347258, −4.998488911442528, −4.246028813817395, −3.647593836576345, −3.000930381570946, −2.400101130004358, −1.748040209043460, −0.7550165433750562, 0, 0.7550165433750562, 1.748040209043460, 2.400101130004358, 3.000930381570946, 3.647593836576345, 4.246028813817395, 4.998488911442528, 5.155430800347258, 6.167992055291974, 6.416923276727690, 7.199254644593773, 7.606105146395709, 8.245266000386979, 8.662542851706668, 8.974590052293777, 9.981896112707722, 10.38326704298579, 10.53340544278546, 11.35422108575262, 11.73295423372671, 12.29589171824976, 12.75197478838588, 13.31215556135623, 13.65365114636965, 14.18634526286317

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