Properties

Degree $2$
Conductor $7581$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 2·5-s − 6-s − 7-s − 3·8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 14-s + 2·15-s − 16-s − 6·17-s + 18-s + 2·20-s + 21-s + 4·22-s + 3·24-s − 25-s + 2·26-s − 27-s + 28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.447·20-s + 0.218·21-s + 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7581\)    =    \(3 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7581} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7581,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8768222459\)
\(L(\frac12)\) \(\approx\) \(0.8768222459\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−7.85302053965317674139971119169, −6.87477517530779045173647189781, −6.42842960047713503559988653252, −5.82904766004805566557712895862, −4.81936947701198622186082780840, −4.36029050874640221146718646433, −3.73298077731310404370734463266, −3.13006637509653258681236844743, −1.71320545529150591858508044629, −0.43525041789359895883868375518, 0.43525041789359895883868375518, 1.71320545529150591858508044629, 3.13006637509653258681236844743, 3.73298077731310404370734463266, 4.36029050874640221146718646433, 4.81936947701198622186082780840, 5.82904766004805566557712895862, 6.42842960047713503559988653252, 6.87477517530779045173647189781, 7.85302053965317674139971119169

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