Properties

Label 2-7800-1.1-c1-0-28
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 − 2.12·7-s + 9-s + 3.15·11-s + 13-s + 0.515·17-s + 6.73·19-s + 2.12·21-s − 4.24·23-s − 27-s + 0.0302·29-s + 7.15·31-s − 3.15·33-s + 5.76·37-s − 39-s − 3.76·41-s − 3.28·43-s − 4.60·47-s − 2.48·49-s − 0.515·51-s + 0.280·53-s − 6.73·57-s + 11.3·59-s − 5.01·61-s − 2.12·63-s + 15.1·67-s + 4.24·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.803·7-s + 0.333·9-s + 0.951·11-s + 0.277·13-s + 0.124·17-s + 1.54·19-s + 0.463·21-s − 0.886·23-s − 0.192·27-s + 0.00562·29-s + 1.28·31-s − 0.549·33-s + 0.947·37-s − 0.160·39-s − 0.587·41-s − 0.500·43-s − 0.672·47-s − 0.354·49-s − 0.0721·51-s + 0.0384·53-s − 0.892·57-s + 1.47·59-s − 0.642·61-s − 0.267·63-s + 1.84·67-s + 0.511·69-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

\(L(1)\) \(\approx\) \(1.585907154\)
\(L(\frac12)\) \(\approx\) \(1.585907154\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.12T + 7T^{2} \)
11 \( 1 - 3.15T + 11T^{2} \)
17 \( 1 - 0.515T + 17T^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 - 0.0302T + 29T^{2} \)
31 \( 1 - 7.15T + 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 + 3.76T + 41T^{2} \)
43 \( 1 + 3.28T + 43T^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 - 0.280T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 5.01T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 2.23T + 71T^{2} \)
73 \( 1 + 2.96T + 73T^{2} \)
79 \( 1 + 8.98T + 79T^{2} \)
83 \( 1 - 5.40T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 - 0.909T + 97T^{2} \)
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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

−7.84765365525870759820662349900, −6.91287890109926689397638981891, −6.52235762616392691878129591883, −5.84555850954056567130427870281, −5.15778848764940520258695333935, −4.25420808248985440632632188487, −3.57445013550573118374694001865, −2.81491138345144566489680797618, −1.56821581285309629810272338961, −0.68108905407635264500246216637, 0.68108905407635264500246216637, 1.56821581285309629810272338961, 2.81491138345144566489680797618, 3.57445013550573118374694001865, 4.25420808248985440632632188487, 5.15778848764940520258695333935, 5.84555850954056567130427870281, 6.52235762616392691878129591883, 6.91287890109926689397638981891, 7.84765365525870759820662349900

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