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Properties

Label 2-23100-1.1-c1-0-13
Degree $2$
Conductor $23100$
Sign $1$
Analytic cond. $184.454$
Root an. cond. $13.5814$
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  − 3-s − 7-s + 9-s − 11-s − 2·13-s + 6·17-s + 8·19-s + 21-s + 6·23-s − 27-s + 6·29-s + 2·31-s + 33-s − 2·37-s + 2·39-s − 8·43-s + 12·47-s + 49-s − 6·51-s − 6·53-s − 8·57-s + 6·59-s + 8·61-s − 63-s − 2·67-s − 6·69-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s − 0.328·37-s + 0.320·39-s − 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 1.05·57-s + 0.781·59-s + 1.02·61-s − 0.125·63-s − 0.244·67-s − 0.722·69-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(184.454\)
Root analytic conductor: \(13.5814\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.075567762\)
\(L(\frac12)\) \(\approx\) \(2.075567762\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.58280952893276, −15.02766943352649, −14.28780009241701, −13.89909483546128, −13.31789504285766, −12.55884004541682, −12.28486786330771, −11.67911298521968, −11.24417725097300, −10.38470757898064, −10.01627584369603, −9.607363307913763, −8.870653343746950, −8.154665193865210, −7.415920901420729, −7.159407941773724, −6.381602339288485, −5.671469433912842, −5.125831504138976, −4.779414010773264, −3.624932220979756, −3.174979659089151, −2.436283092613726, −1.213140050061292, −0.6997767158769507, 0.6997767158769507, 1.213140050061292, 2.436283092613726, 3.174979659089151, 3.624932220979756, 4.779414010773264, 5.125831504138976, 5.671469433912842, 6.381602339288485, 7.159407941773724, 7.415920901420729, 8.154665193865210, 8.870653343746950, 9.607363307913763, 10.01627584369603, 10.38470757898064, 11.24417725097300, 11.67911298521968, 12.28486786330771, 12.55884004541682, 13.31789504285766, 13.89909483546128, 14.28780009241701, 15.02766943352649, 15.58280952893276

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