Properties

Label 2-7800-1.1-c1-0-19
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 − 3.32·7-s + 9-s + 4.84·11-s − 13-s + 1.11·17-s + 4.05·19-s + 3.32·21-s + 4.16·23-s − 27-s + 5.64·29-s − 7.49·31-s − 4.84·33-s + 8.69·37-s + 39-s + 5.63·41-s − 12.8·43-s + 6.89·47-s + 4.05·49-s − 1.11·51-s − 12.2·53-s − 4.05·57-s + 0.910·59-s + 2.46·61-s − 3.32·63-s − 10.0·67-s − 4.16·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.25·7-s + 0.333·9-s + 1.46·11-s − 0.277·13-s + 0.271·17-s + 0.929·19-s + 0.725·21-s + 0.869·23-s − 0.192·27-s + 1.04·29-s − 1.34·31-s − 0.843·33-s + 1.43·37-s + 0.160·39-s + 0.880·41-s − 1.95·43-s + 1.00·47-s + 0.578·49-s − 0.156·51-s − 1.67·53-s − 0.536·57-s + 0.118·59-s + 0.316·61-s − 0.418·63-s − 1.22·67-s − 0.501·69-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

\(L(1)\) \(\approx\) \(1.473832100\)
\(L(\frac12)\) \(\approx\) \(1.473832100\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.32T + 7T^{2} \)
11 \( 1 - 4.84T + 11T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 - 4.05T + 19T^{2} \)
23 \( 1 - 4.16T + 23T^{2} \)
29 \( 1 - 5.64T + 29T^{2} \)
31 \( 1 + 7.49T + 31T^{2} \)
37 \( 1 - 8.69T + 37T^{2} \)
41 \( 1 - 5.63T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 - 6.89T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 - 0.910T + 59T^{2} \)
61 \( 1 - 2.46T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 8.05T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 2.67T + 83T^{2} \)
89 \( 1 + 3.68T + 89T^{2} \)
97 \( 1 - 6.64T + 97T^{2} \)
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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

−7.57819474508497242994523852977, −7.09142974059603628381585962518, −6.36406421130495861499625867080, −6.00295293551649515778099321408, −5.07556610254713798432033766266, −4.31203436199420879128053259538, −3.46758675196988084927849899090, −2.90681412948234669084204512903, −1.55464762127245655000248891960, −0.65542749639826579741581481344, 0.65542749639826579741581481344, 1.55464762127245655000248891960, 2.90681412948234669084204512903, 3.46758675196988084927849899090, 4.31203436199420879128053259538, 5.07556610254713798432033766266, 6.00295293551649515778099321408, 6.36406421130495861499625867080, 7.09142974059603628381585962518, 7.57819474508497242994523852977

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