Properties

Label 2-29400-1.1-c1-0-113
Degree $2$
Conductor $29400$
Sign $-1$
Analytic cond. $234.760$
Root an. cond. $15.3218$
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  + 3-s + 9-s − 13-s + 6·17-s − 19-s + 2·23-s + 27-s − 3·37-s − 39-s − 8·43-s − 2·47-s + 6·51-s − 2·53-s − 57-s − 10·59-s − 5·61-s + 3·67-s + 2·69-s − 12·71-s − 13·73-s + 11·79-s + 81-s − 2·83-s − 6·89-s − 97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.277·13-s + 1.45·17-s − 0.229·19-s + 0.417·23-s + 0.192·27-s − 0.493·37-s − 0.160·39-s − 1.21·43-s − 0.291·47-s + 0.840·51-s − 0.274·53-s − 0.132·57-s − 1.30·59-s − 0.640·61-s + 0.366·67-s + 0.240·69-s − 1.42·71-s − 1.52·73-s + 1.23·79-s + 1/9·81-s − 0.219·83-s − 0.635·89-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(234.760\)
Root analytic conductor: \(15.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−15.31281797391417, −14.82584392912098, −14.43483816519155, −13.90054036811731, −13.32025880240404, −12.86896585232964, −12.07724437739580, −11.98945816989429, −11.06374189204033, −10.51026797544434, −9.992300128622407, −9.484573893465396, −8.921007126952533, −8.279810029017825, −7.805712012401046, −7.257368335624750, −6.647806542599769, −5.934244014434702, −5.281757824288377, −4.688325247263880, −3.955880018115057, −3.204039573237367, −2.841880191864047, −1.805051216043333, −1.219423791828760, 0, 1.219423791828760, 1.805051216043333, 2.841880191864047, 3.204039573237367, 3.955880018115057, 4.688325247263880, 5.281757824288377, 5.934244014434702, 6.647806542599769, 7.257368335624750, 7.805712012401046, 8.279810029017825, 8.921007126952533, 9.484573893465396, 9.992300128622407, 10.51026797544434, 11.06374189204033, 11.98945816989429, 12.07724437739580, 12.86896585232964, 13.32025880240404, 13.90054036811731, 14.43483816519155, 14.82584392912098, 15.31281797391417

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