Properties

Label 2-7800-1.1-c1-0-51
Degree $2$
Conductor $7800$
Sign $1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
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 + 4.92·7-s + 9-s − 1.38·11-s − 13-s − 0.195·17-s + 4.92·21-s + 2.19·23-s + 27-s + 7.49·29-s − 1.38·33-s − 6.05·37-s − 39-s − 2.11·41-s − 3.49·43-s − 2.31·47-s + 17.2·49-s − 0.195·51-s + 6.05·53-s + 9.43·59-s − 3.69·61-s + 4.92·63-s + 12.6·67-s + 2.19·69-s + 13.9·71-s − 4.73·73-s − 6.81·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.86·7-s + 0.333·9-s − 0.417·11-s − 0.277·13-s − 0.0473·17-s + 1.07·21-s + 0.457·23-s + 0.192·27-s + 1.39·29-s − 0.240·33-s − 0.994·37-s − 0.160·39-s − 0.330·41-s − 0.533·43-s − 0.337·47-s + 2.46·49-s − 0.0273·51-s + 0.831·53-s + 1.22·59-s − 0.473·61-s + 0.620·63-s + 1.54·67-s + 0.264·69-s + 1.65·71-s − 0.553·73-s − 0.776·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.395410097\)
\(L(\frac12)\) \(\approx\) \(3.395410097\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 4.92T + 7T^{2} \)
11 \( 1 + 1.38T + 11T^{2} \)
17 \( 1 + 0.195T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 - 7.49T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6.05T + 37T^{2} \)
41 \( 1 + 2.11T + 41T^{2} \)
43 \( 1 + 3.49T + 43T^{2} \)
47 \( 1 + 2.31T + 47T^{2} \)
53 \( 1 - 6.05T + 53T^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 + 3.69T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 4.73T + 73T^{2} \)
79 \( 1 + 8.07T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 3.02T + 89T^{2} \)
97 \( 1 + 6.01T + 97T^{2} \)
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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

−8.186596539982549977663567060070, −7.24012532903157890762542665052, −6.71341870777901671302674281551, −5.50116811892375664339324999793, −4.99675035837101901967681012314, −4.42897146406942717021275356424, −3.52014309589358310634831340341, −2.52370258772697067496418962887, −1.87755573750994102547755197948, −0.936028014058307779117766079293, 0.936028014058307779117766079293, 1.87755573750994102547755197948, 2.52370258772697067496418962887, 3.52014309589358310634831340341, 4.42897146406942717021275356424, 4.99675035837101901967681012314, 5.50116811892375664339324999793, 6.71341870777901671302674281551, 7.24012532903157890762542665052, 8.186596539982549977663567060070

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