Properties

Label 2-6900-1.1-c1-0-1
Degree $2$
Conductor $6900$
Sign $1$
Analytic cond. $55.0967$
Root an. cond. $7.42272$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.52·7-s + 9-s − 3.59·11-s − 5.59·13-s − 4.07·17-s − 1.59·19-s + 1.52·21-s − 23-s − 27-s − 4.07·29-s + 2.47·31-s + 3.59·33-s − 9.66·37-s + 5.59·39-s + 5.01·41-s − 7.05·43-s + 4.54·47-s − 4.66·49-s + 4.07·51-s − 5.01·53-s + 1.59·57-s + 4.07·59-s + 6.54·61-s − 1.52·63-s − 2.47·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.576·7-s + 0.333·9-s − 1.08·11-s − 1.55·13-s − 0.987·17-s − 0.366·19-s + 0.333·21-s − 0.208·23-s − 0.192·27-s − 0.756·29-s + 0.444·31-s + 0.626·33-s − 1.58·37-s + 0.896·39-s + 0.783·41-s − 1.07·43-s + 0.663·47-s − 0.667·49-s + 0.570·51-s − 0.689·53-s + 0.211·57-s + 0.530·59-s + 0.838·61-s − 0.192·63-s − 0.302·67-s + 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(55.0967\)
Root analytic conductor: \(7.42272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3934841631\)
\(L(\frac12)\) \(\approx\) \(0.3934841631\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 + T \)
good7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 + 3.59T + 11T^{2} \)
13 \( 1 + 5.59T + 13T^{2} \)
17 \( 1 + 4.07T + 17T^{2} \)
19 \( 1 + 1.59T + 19T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 + 9.66T + 37T^{2} \)
41 \( 1 - 5.01T + 41T^{2} \)
43 \( 1 + 7.05T + 43T^{2} \)
47 \( 1 - 4.54T + 47T^{2} \)
53 \( 1 + 5.01T + 53T^{2} \)
59 \( 1 - 4.07T + 59T^{2} \)
61 \( 1 - 6.54T + 61T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 + 5.01T + 71T^{2} \)
73 \( 1 - 8.79T + 73T^{2} \)
79 \( 1 - 2.94T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.83741547722117823150346250081, −7.15305321633665806510064267193, −6.64537206269279963816912058935, −5.80987188854446661154885520550, −5.07759117152315213829401336276, −4.60095532205306933196083328365, −3.58578129139409391289793983171, −2.61871749299982898473656769770, −1.95162090301940564751516425685, −0.30669865691687520325067517659, 0.30669865691687520325067517659, 1.95162090301940564751516425685, 2.61871749299982898473656769770, 3.58578129139409391289793983171, 4.60095532205306933196083328365, 5.07759117152315213829401336276, 5.80987188854446661154885520550, 6.64537206269279963816912058935, 7.15305321633665806510064267193, 7.83741547722117823150346250081

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