Properties

Label 2-308550-1.1-c1-0-4
Degree $2$
Conductor $308550$
Sign $1$
Analytic cond. $2463.78$
Root an. cond. $49.6365$
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  − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s + 2·13-s + 4·14-s + 16-s − 17-s − 18-s + 4·19-s + 4·21-s + 24-s − 2·26-s − 27-s − 4·28-s + 6·29-s + 8·31-s − 32-s + 34-s + 36-s + 10·37-s − 4·38-s − 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(2463.78\)
Root analytic conductor: \(49.6365\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{308550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 308550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2885947697\)
\(L(\frac12)\) \(\approx\) \(0.2885947697\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
11 \( 1 \)
17 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.69400343725032, −12.04849430639541, −11.58896797850063, −11.48035136752343, −10.72184729395197, −10.18995308632263, −9.934057778549646, −9.637404849134472, −9.071647629942965, −8.506298545161316, −8.115556545566336, −7.563229071063593, −6.886129811367520, −6.577272974873611, −6.295286966343102, −5.767067524078785, −5.190839731265652, −4.541656004584977, −4.023691386322010, −3.275074670661814, −2.932063530617283, −2.470729236992899, −1.321575125069958, −1.169926536596563, −0.1844792744605169, 0.1844792744605169, 1.169926536596563, 1.321575125069958, 2.470729236992899, 2.932063530617283, 3.275074670661814, 4.023691386322010, 4.541656004584977, 5.190839731265652, 5.767067524078785, 6.295286966343102, 6.577272974873611, 6.886129811367520, 7.563229071063593, 8.115556545566336, 8.506298545161316, 9.071647629942965, 9.637404849134472, 9.934057778549646, 10.18995308632263, 10.72184729395197, 11.48035136752343, 11.58896797850063, 12.04849430639541, 12.69400343725032

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