Properties

Label 2-7800-1.1-c1-0-108
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.70·7-s + 9-s − 5.70·11-s + 13-s − 5.70·17-s − 2·19-s + 3.70·21-s − 0.298·23-s + 27-s − 3.40·29-s + 3.40·31-s − 5.70·33-s + 0.298·37-s + 39-s + 4.29·41-s − 4·43-s − 11.4·47-s + 6.70·49-s − 5.70·51-s − 4.29·53-s − 2·57-s + 10.8·59-s − 3.70·61-s + 3.70·63-s + 4·67-s − 0.298·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.39·7-s + 0.333·9-s − 1.71·11-s + 0.277·13-s − 1.38·17-s − 0.458·19-s + 0.807·21-s − 0.0622·23-s + 0.192·27-s − 0.631·29-s + 0.611·31-s − 0.992·33-s + 0.0490·37-s + 0.160·39-s + 0.671·41-s − 0.609·43-s − 1.66·47-s + 0.957·49-s − 0.798·51-s − 0.590·53-s − 0.264·57-s + 1.40·59-s − 0.473·61-s + 0.466·63-s + 0.488·67-s − 0.0359·69-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: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.70T + 7T^{2} \)
11 \( 1 + 5.70T + 11T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 0.298T + 23T^{2} \)
29 \( 1 + 3.40T + 29T^{2} \)
31 \( 1 - 3.40T + 31T^{2} \)
37 \( 1 - 0.298T + 37T^{2} \)
41 \( 1 - 4.29T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 4.29T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 3.70T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 + 0.596T + 73T^{2} \)
79 \( 1 + 1.70T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 - 1.10T + 97T^{2} \)
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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

−7.67785063904964523830487751811, −7.00720693043923507696612725962, −6.10187293409555604159188222360, −5.20913974332270176626736752648, −4.71053643156127050637246655074, −4.04825501854536297769230575033, −2.90866040352754123176957515802, −2.25772694655059107006728264401, −1.52289531914570518966651386584, 0, 1.52289531914570518966651386584, 2.25772694655059107006728264401, 2.90866040352754123176957515802, 4.04825501854536297769230575033, 4.71053643156127050637246655074, 5.20913974332270176626736752648, 6.10187293409555604159188222360, 7.00720693043923507696612725962, 7.67785063904964523830487751811

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