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Properties

Label 2-221760-1.1-c1-0-170
Degree $2$
Conductor $221760$
Sign $-1$
Analytic cond. $1770.76$
Root an. cond. $42.0804$
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 − 7-s + 11-s − 6·13-s − 2·17-s − 4·19-s + 25-s − 10·29-s + 8·31-s − 35-s − 6·37-s − 10·41-s + 4·43-s + 8·47-s + 49-s + 6·53-s + 55-s + 12·59-s − 6·61-s − 6·65-s − 4·67-s + 16·71-s + 10·73-s − 77-s − 12·83-s − 2·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.301·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.85·29-s + 1.43·31-s − 0.169·35-s − 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 1.56·59-s − 0.768·61-s − 0.744·65-s − 0.488·67-s + 1.89·71-s + 1.17·73-s − 0.113·77-s − 1.31·83-s − 0.216·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221760\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(1770.76\)
Root analytic conductor: \(42.0804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.29425233945484, −12.61693447512597, −12.28463367240332, −11.99229147907911, −11.27341602856153, −10.89014170946927, −10.21378388123410, −10.02605850063171, −9.399614062664037, −9.154386944872857, −8.433129559957577, −8.142907820599964, −7.274695893182005, −7.011586865538150, −6.646369697588952, −5.931109816223893, −5.501152160118870, −4.969653335872793, −4.445896018116487, −3.883881584167566, −3.333058776055350, −2.470925060258729, −2.281790588122516, −1.638943947049900, −0.6673144261705465, 0, 0.6673144261705465, 1.638943947049900, 2.281790588122516, 2.470925060258729, 3.333058776055350, 3.883881584167566, 4.445896018116487, 4.969653335872793, 5.501152160118870, 5.931109816223893, 6.646369697588952, 7.011586865538150, 7.274695893182005, 8.142907820599964, 8.433129559957577, 9.154386944872857, 9.399614062664037, 10.02605850063171, 10.21378388123410, 10.89014170946927, 11.27341602856153, 11.99229147907911, 12.28463367240332, 12.61693447512597, 13.29425233945484

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