Properties

Label 2-7800-1.1-c1-0-29
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 − 2.13·7-s + 9-s − 3.02·11-s + 13-s − 1.62·17-s + 7.08·19-s − 2.13·21-s + 3.62·23-s + 27-s − 2.46·29-s − 8.37·31-s − 3.02·33-s + 2.65·37-s + 39-s + 4.19·41-s + 4.76·43-s − 7.11·47-s − 2.46·49-s − 1.62·51-s + 7.58·53-s + 7.08·57-s + 14.6·59-s − 14.0·61-s − 2.13·63-s + 1.01·67-s + 3.62·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.805·7-s + 0.333·9-s − 0.912·11-s + 0.277·13-s − 0.393·17-s + 1.62·19-s − 0.464·21-s + 0.755·23-s + 0.192·27-s − 0.457·29-s − 1.50·31-s − 0.526·33-s + 0.436·37-s + 0.160·39-s + 0.655·41-s + 0.726·43-s − 1.03·47-s − 0.351·49-s − 0.227·51-s + 1.04·53-s + 0.938·57-s + 1.90·59-s − 1.79·61-s − 0.268·63-s + 0.123·67-s + 0.436·69-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

\(L(1)\) \(\approx\) \(2.055888987\)
\(L(\frac12)\) \(\approx\) \(2.055888987\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.13T + 7T^{2} \)
11 \( 1 + 3.02T + 11T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 - 7.08T + 19T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + 8.37T + 31T^{2} \)
37 \( 1 - 2.65T + 37T^{2} \)
41 \( 1 - 4.19T + 41T^{2} \)
43 \( 1 - 4.76T + 43T^{2} \)
47 \( 1 + 7.11T + 47T^{2} \)
53 \( 1 - 7.58T + 53T^{2} \)
59 \( 1 - 14.6T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 1.01T + 67T^{2} \)
71 \( 1 - 9.32T + 71T^{2} \)
73 \( 1 - 2.34T + 73T^{2} \)
79 \( 1 + 7.87T + 79T^{2} \)
83 \( 1 - 0.396T + 83T^{2} \)
89 \( 1 - 6.52T + 89T^{2} \)
97 \( 1 + 0.794T + 97T^{2} \)
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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

−7.72297094938744737221373578519, −7.31372395106922088722067785137, −6.59636145473823802041486503893, −5.66179686459640340967112029825, −5.17549359045505895476272952567, −4.15719934003535503809530744661, −3.34178717156770613489131993537, −2.85877761733018880994193892277, −1.90097748573005664856536390881, −0.68131317992761279506823685965, 0.68131317992761279506823685965, 1.90097748573005664856536390881, 2.85877761733018880994193892277, 3.34178717156770613489131993537, 4.15719934003535503809530744661, 5.17549359045505895476272952567, 5.66179686459640340967112029825, 6.59636145473823802041486503893, 7.31372395106922088722067785137, 7.72297094938744737221373578519

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