Properties

Label 2-100800-1.1-c1-0-171
Degree $2$
Conductor $100800$
Sign $-1$
Analytic cond. $804.892$
Root an. cond. $28.3706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 6·13-s + 2·17-s + 6·29-s − 8·31-s − 10·37-s − 2·41-s − 4·43-s − 8·47-s + 49-s + 2·53-s + 8·59-s + 14·61-s + 12·67-s − 16·71-s − 2·73-s + 4·77-s + 8·79-s + 8·83-s − 10·89-s + 6·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.274·53-s + 1.04·59-s + 1.79·61-s + 1.46·67-s − 1.89·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s + 0.878·83-s − 1.05·89-s + 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(804.892\)
Root analytic conductor: \(28.3706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.05413105707264, −13.38819287227690, −12.99190291196859, −12.52860115445543, −12.14346819688546, −11.61541591257306, −11.05211757581408, −10.28325269831507, −10.13796078762018, −9.756410442213990, −9.014303799450845, −8.503798932292613, −7.949524993358179, −7.487146094781965, −6.878233161411640, −6.661723425327035, −5.566150242221185, −5.343341702820986, −4.903576252916919, −4.200689543641700, −3.373765011156887, −3.018779922786098, −2.256722815081749, −1.837507404427671, −0.6599961150823585, 0, 0.6599961150823585, 1.837507404427671, 2.256722815081749, 3.018779922786098, 3.373765011156887, 4.200689543641700, 4.903576252916919, 5.343341702820986, 5.566150242221185, 6.661723425327035, 6.878233161411640, 7.487146094781965, 7.949524993358179, 8.503798932292613, 9.014303799450845, 9.756410442213990, 10.13796078762018, 10.28325269831507, 11.05211757581408, 11.61541591257306, 12.14346819688546, 12.52860115445543, 12.99190291196859, 13.38819287227690, 14.05413105707264

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