Properties

Label 2-68400-1.1-c1-0-27
Degree $2$
Conductor $68400$
Sign $1$
Analytic cond. $546.176$
Root an. cond. $23.3704$
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  + 4·11-s − 2·13-s + 2·17-s + 19-s − 4·23-s − 6·29-s − 4·31-s + 6·37-s − 10·41-s − 4·43-s + 12·47-s − 7·49-s + 6·53-s − 12·59-s − 2·61-s + 4·67-s + 8·71-s + 6·73-s + 4·79-s + 12·83-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 0.834·23-s − 1.11·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.75·47-s − 49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.450·79-s + 1.31·83-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(546.176\)
Root analytic conductor: \(23.3704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{68400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 68400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.975655532\)
\(L(\frac12)\) \(\approx\) \(1.975655532\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.26106281404404, −13.65210538864199, −13.29236737722763, −12.46211191923838, −12.18104919073839, −11.77512163370166, −11.13232942908786, −10.73437220553920, −9.953801531973658, −9.598711560146575, −9.173055688800047, −8.598706661743850, −7.820852796315665, −7.620875595075042, −6.728898293640096, −6.535105106329721, −5.662828202249695, −5.335835468557689, −4.551468622625998, −3.901545955256834, −3.548580591806693, −2.735727427957525, −1.937613148541180, −1.421791280928730, −0.4712273274452571, 0.4712273274452571, 1.421791280928730, 1.937613148541180, 2.735727427957525, 3.548580591806693, 3.901545955256834, 4.551468622625998, 5.335835468557689, 5.662828202249695, 6.535105106329721, 6.728898293640096, 7.620875595075042, 7.820852796315665, 8.598706661743850, 9.173055688800047, 9.598711560146575, 9.953801531973658, 10.73437220553920, 11.13232942908786, 11.77512163370166, 12.18104919073839, 12.46211191923838, 13.29236737722763, 13.65210538864199, 14.26106281404404

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