Properties

Label 2-100800-1.1-c1-0-397
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 3·13-s − 7·17-s − 6·19-s + 9·23-s + 3·29-s + 7·31-s − 10·37-s + 41-s − 13·43-s − 2·47-s + 49-s − 53-s + 11·59-s − 13·61-s + 8·71-s − 8·73-s + 4·77-s − 4·79-s − 7·83-s + 14·89-s + 3·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 0.832·13-s − 1.69·17-s − 1.37·19-s + 1.87·23-s + 0.557·29-s + 1.25·31-s − 1.64·37-s + 0.156·41-s − 1.98·43-s − 0.291·47-s + 1/7·49-s − 0.137·53-s + 1.43·59-s − 1.66·61-s + 0.949·71-s − 0.936·73-s + 0.455·77-s − 0.450·79-s − 0.768·83-s + 1.48·89-s + 0.314·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: $\chi_{100800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.18829875735127, −13.58585443255413, −13.08918409977476, −13.01881295573983, −12.35697415974542, −11.80712859937854, −11.25708975973939, −10.66881836533359, −10.39610674257915, −9.937070850535708, −9.203438533848656, −8.694567522088246, −8.426976100796897, −7.750316718943380, −7.072162377025892, −6.614624416038137, −6.426277261800744, −5.394153683252380, −4.931770714989901, −4.651545901128890, −3.904427656537450, −3.063102361524481, −2.628702647516007, −2.154754474374887, −1.273766063213769, 0, 0, 1.273766063213769, 2.154754474374887, 2.628702647516007, 3.063102361524481, 3.904427656537450, 4.651545901128890, 4.931770714989901, 5.394153683252380, 6.426277261800744, 6.614624416038137, 7.072162377025892, 7.750316718943380, 8.426976100796897, 8.694567522088246, 9.203438533848656, 9.937070850535708, 10.39610674257915, 10.66881836533359, 11.25708975973939, 11.80712859937854, 12.35697415974542, 13.01881295573983, 13.08918409977476, 13.58585443255413, 14.18829875735127

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