Properties

Label 2-7800-1.1-c1-0-21
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 + 4.41·7-s + 9-s − 4.41·11-s − 13-s − 1.31·17-s + 0.311·19-s − 4.41·21-s + 1.37·23-s − 27-s + 29-s − 4.41·31-s + 4.41·33-s − 3.68·37-s + 39-s + 1.68·41-s + 2·43-s + 2.73·47-s + 12.5·49-s + 1.31·51-s − 2.37·53-s − 0.311·57-s − 1.04·59-s − 4.06·61-s + 4.41·63-s + 6.10·67-s − 1.37·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.67·7-s + 0.333·9-s − 1.33·11-s − 0.277·13-s − 0.318·17-s + 0.0714·19-s − 0.964·21-s + 0.287·23-s − 0.192·27-s + 0.185·29-s − 0.793·31-s + 0.769·33-s − 0.606·37-s + 0.160·39-s + 0.263·41-s + 0.304·43-s + 0.398·47-s + 1.78·49-s + 0.183·51-s − 0.326·53-s − 0.0412·57-s − 0.135·59-s − 0.520·61-s + 0.556·63-s + 0.746·67-s − 0.165·69-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

\(L(1)\) \(\approx\) \(1.683173669\)
\(L(\frac12)\) \(\approx\) \(1.683173669\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.41T + 7T^{2} \)
11 \( 1 + 4.41T + 11T^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 - 0.311T + 19T^{2} \)
23 \( 1 - 1.37T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + 4.41T + 31T^{2} \)
37 \( 1 + 3.68T + 37T^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 2.73T + 47T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 + 1.04T + 59T^{2} \)
61 \( 1 + 4.06T + 61T^{2} \)
67 \( 1 - 6.10T + 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 5.04T + 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 + 5.37T + 97T^{2} \)
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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

−7.73585171039311923190884504774, −7.38355632244138927315037595730, −6.44114388191899731309854616889, −5.52725678080984919635680141102, −5.05348836243339910270376436089, −4.62521586558471810135190829579, −3.62777619991229592778537906290, −2.45170719434205755002081763361, −1.80965145413848804474142019855, −0.66631521545458343069114498580, 0.66631521545458343069114498580, 1.80965145413848804474142019855, 2.45170719434205755002081763361, 3.62777619991229592778537906290, 4.62521586558471810135190829579, 5.05348836243339910270376436089, 5.52725678080984919635680141102, 6.44114388191899731309854616889, 7.38355632244138927315037595730, 7.73585171039311923190884504774

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