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Properties

Label 2-121680-1.1-c1-0-31
Degree $2$
Conductor $121680$
Sign $1$
Analytic cond. $971.619$
Root an. cond. $31.1708$
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  + 5-s − 4·11-s + 6·17-s + 4·19-s + 8·23-s + 25-s − 6·29-s − 8·31-s + 10·37-s − 6·41-s − 4·43-s − 7·49-s + 10·53-s − 4·55-s − 4·59-s − 2·61-s − 12·67-s − 16·71-s − 2·73-s + 16·79-s + 12·83-s + 6·85-s + 10·89-s + 4·95-s + 6·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s + 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.937·41-s − 0.609·43-s − 49-s + 1.37·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 1.46·67-s − 1.89·71-s − 0.234·73-s + 1.80·79-s + 1.31·83-s + 0.650·85-s + 1.05·89-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121680\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(971.619\)
Root analytic conductor: \(31.1708\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 121680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.510588582\)
\(L(\frac12)\) \(\approx\) \(2.510588582\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.31234851343572, −13.19650628347277, −12.78556839643630, −12.00179387644391, −11.74119571479228, −10.97761103920480, −10.69811199254371, −10.13722533020575, −9.678296285175896, −9.152313964588066, −8.826851031320614, −7.830989481537434, −7.730299617046435, −7.251604644699247, −6.569484408221284, −5.860467717418727, −5.421053825218351, −5.132101551870358, −4.527893379861070, −3.544808617196445, −3.209267985564961, −2.684525119290806, −1.872690829654285, −1.266591940588855, −0.4993807997461223, 0.4993807997461223, 1.266591940588855, 1.872690829654285, 2.684525119290806, 3.209267985564961, 3.544808617196445, 4.527893379861070, 5.132101551870358, 5.421053825218351, 5.860467717418727, 6.569484408221284, 7.251604644699247, 7.730299617046435, 7.830989481537434, 8.826851031320614, 9.152313964588066, 9.678296285175896, 10.13722533020575, 10.69811199254371, 10.97761103920480, 11.74119571479228, 12.00179387644391, 12.78556839643630, 13.19650628347277, 13.31234851343572

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