Properties

Label 2-107712-1.1-c1-0-26
Degree $2$
Conductor $107712$
Sign $1$
Analytic cond. $860.084$
Root an. cond. $29.3271$
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  + 2·5-s + 2·7-s + 11-s − 4·13-s − 17-s + 2·19-s + 2·23-s − 25-s + 2·29-s + 4·31-s + 4·35-s − 6·37-s − 6·41-s + 2·43-s − 3·49-s − 12·53-s + 2·55-s + 14·59-s − 6·61-s − 8·65-s − 4·67-s − 2·71-s − 8·73-s + 2·77-s + 2·79-s + 12·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.301·11-s − 1.10·13-s − 0.242·17-s + 0.458·19-s + 0.417·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s + 0.304·43-s − 3/7·49-s − 1.64·53-s + 0.269·55-s + 1.82·59-s − 0.768·61-s − 0.992·65-s − 0.488·67-s − 0.237·71-s − 0.936·73-s + 0.227·77-s + 0.225·79-s + 1.31·83-s − 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(107712\)    =    \(2^{6} \cdot 3^{2} \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(860.084\)
Root analytic conductor: \(29.3271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 107712,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.821837440\)
\(L(\frac12)\) \(\approx\) \(2.821837440\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.84884706738042, −13.22768349139453, −12.75673252855428, −12.14174637631999, −11.74230575937612, −11.35303275843665, −10.62873598909213, −10.21850962207738, −9.777657786317019, −9.288652336235593, −8.806736848700054, −8.204896369190435, −7.705557463213623, −7.165571105212749, −6.578572476807736, −6.174645611637261, −5.339351170116017, −5.107665861808469, −4.572548978873594, −3.903399022742443, −3.065270873997956, −2.598399107346486, −1.796762627955065, −1.507806155664351, −0.5025583542803073, 0.5025583542803073, 1.507806155664351, 1.796762627955065, 2.598399107346486, 3.065270873997956, 3.903399022742443, 4.572548978873594, 5.107665861808469, 5.339351170116017, 6.174645611637261, 6.578572476807736, 7.165571105212749, 7.705557463213623, 8.204896369190435, 8.806736848700054, 9.288652336235593, 9.777657786317019, 10.21850962207738, 10.62873598909213, 11.35303275843665, 11.74230575937612, 12.14174637631999, 12.75673252855428, 13.22768349139453, 13.84884706738042

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