Properties

Label 2-7800-1.1-c1-0-53
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

\(L(1)\) \(\approx\) \(3.432342315\)
\(L(\frac12)\) \(\approx\) \(3.432342315\)
\(L(\frac{3}{2})\) 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

−7.79143995920375914093072240097, −7.27187203208441415433761563586, −6.64830092603914128731069642449, −5.69828887806746847553125071080, −4.93953817050152635471610312513, −4.34253535925728150234271408013, −3.43796396340699461878803761398, −2.76464827656892912960537622508, −1.69822645751751309983858918802, −0.980656175413720460102212646079, 0.980656175413720460102212646079, 1.69822645751751309983858918802, 2.76464827656892912960537622508, 3.43796396340699461878803761398, 4.34253535925728150234271408013, 4.93953817050152635471610312513, 5.69828887806746847553125071080, 6.64830092603914128731069642449, 7.27187203208441415433761563586, 7.79143995920375914093072240097

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