Properties

Label 2-297024-1.1-c1-0-48
Degree $2$
Conductor $297024$
Sign $1$
Analytic cond. $2371.74$
Root an. cond. $48.7006$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4·5-s − 7-s + 9-s − 4·11-s − 13-s + 4·15-s + 17-s − 21-s + 4·23-s + 11·25-s + 27-s + 4·29-s − 8·31-s − 4·33-s − 4·35-s − 4·37-s − 39-s − 4·43-s + 4·45-s − 10·47-s + 49-s + 51-s − 6·53-s − 16·55-s − 2·59-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 1.03·15-s + 0.242·17-s − 0.218·21-s + 0.834·23-s + 11/5·25-s + 0.192·27-s + 0.742·29-s − 1.43·31-s − 0.696·33-s − 0.676·35-s − 0.657·37-s − 0.160·39-s − 0.609·43-s + 0.596·45-s − 1.45·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 2.15·55-s − 0.260·59-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297024\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(2371.74\)
Root analytic conductor: \(48.7006\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{297024} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 297024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.654792543\)
\(L(\frac12)\) \(\approx\) \(3.654792543\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.80016915656756, −12.55927755874740, −11.89745403728698, −11.08578009701768, −10.69520764808093, −10.38440163768568, −9.816129368787087, −9.525864024892191, −9.178990512212781, −8.618269724672375, −8.065453596322575, −7.683424222207219, −6.903391085202451, −6.644638504243581, −6.188802950672914, −5.343703191345581, −5.234275141855318, −4.880557577742570, −3.904723288513059, −3.325328600407045, −2.745267080348975, −2.493784033109198, −1.768092813368974, −1.407579964782767, −0.4510414992356300, 0.4510414992356300, 1.407579964782767, 1.768092813368974, 2.493784033109198, 2.745267080348975, 3.325328600407045, 3.904723288513059, 4.880557577742570, 5.234275141855318, 5.343703191345581, 6.188802950672914, 6.644638504243581, 6.903391085202451, 7.683424222207219, 8.065453596322575, 8.618269724672375, 9.178990512212781, 9.525864024892191, 9.816129368787087, 10.38440163768568, 10.69520764808093, 11.08578009701768, 11.89745403728698, 12.55927755874740, 12.80016915656756

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