Properties

Label 2-7800-1.1-c1-0-109
Degree $2$
Conductor $7800$
Sign $-1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s + 3·11-s − 13-s − 7·17-s + 21-s − 7·23-s + 27-s − 4·29-s + 8·31-s + 3·33-s − 5·37-s − 39-s − 3·41-s − 8·43-s + 6·47-s − 6·49-s − 7·51-s − 11·53-s − 4·59-s + 61-s + 63-s + 12·67-s − 7·69-s − 9·71-s + 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s + 0.218·21-s − 1.45·23-s + 0.192·27-s − 0.742·29-s + 1.43·31-s + 0.522·33-s − 0.821·37-s − 0.160·39-s − 0.468·41-s − 1.21·43-s + 0.875·47-s − 6/7·49-s − 0.980·51-s − 1.51·53-s − 0.520·59-s + 0.128·61-s + 0.125·63-s + 1.46·67-s − 0.842·69-s − 1.06·71-s + 0.702·73-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: yes
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 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−7.61744070848064466249311358956, −6.66829151979091858079197916188, −6.43068742573207513663124904351, −5.32764273551975483527464339315, −4.47201915511794813566464693222, −4.03379937936608528946586711485, −3.09993684713890123852628980438, −2.13820228431270433295973167861, −1.53595124430068697367324888802, 0, 1.53595124430068697367324888802, 2.13820228431270433295973167861, 3.09993684713890123852628980438, 4.03379937936608528946586711485, 4.47201915511794813566464693222, 5.32764273551975483527464339315, 6.43068742573207513663124904351, 6.66829151979091858079197916188, 7.61744070848064466249311358956

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