Properties

Label 2-79560-1.1-c1-0-19
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 + 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

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

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 \( 1 + p T^{2} \)
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

−14.38462505627754, −13.69595712018216, −13.10130812136605, −12.82616264008067, −12.41846382579255, −11.68837354938100, −11.13608083930454, −10.89424618028461, −10.31119399073796, −9.737038316211794, −9.266734856840203, −8.525242244309437, −8.072279230102705, −7.858258830031911, −7.062525048250635, −6.532664524916063, −6.060091880175916, −5.297919912090039, −4.880605989350359, −4.227441091231079, −3.719135407844748, −2.906395895825885, −2.476578318086874, −1.718667411858223, −0.7640419578366023, 0, 0.7640419578366023, 1.718667411858223, 2.476578318086874, 2.906395895825885, 3.719135407844748, 4.227441091231079, 4.880605989350359, 5.297919912090039, 6.060091880175916, 6.532664524916063, 7.062525048250635, 7.858258830031911, 8.072279230102705, 8.525242244309437, 9.266734856840203, 9.737038316211794, 10.31119399073796, 10.89424618028461, 11.13608083930454, 11.68837354938100, 12.41846382579255, 12.82616264008067, 13.10130812136605, 13.69595712018216, 14.38462505627754

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