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Properties

Label 2-303450-1.1-c1-0-38
Degree $2$
Conductor $303450$
Sign $1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
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 − 7-s − 8-s + 9-s + 3·11-s − 12-s + 2·13-s + 14-s + 16-s − 18-s − 19-s + 21-s − 3·22-s + 6·23-s + 24-s − 2·26-s − 27-s − 28-s + 3·29-s − 5·31-s − 32-s − 3·33-s + 36-s + 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.218·21-s − 0.639·22-s + 1.25·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s − 0.898·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.515844151\)
\(L(\frac12)\) \(\approx\) \(1.515844151\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−12.49365460409229, −12.23771573727638, −11.68014515184052, −11.10152720049267, −10.85564192093186, −10.62273250455371, −9.750990547232765, −9.465423549975462, −9.055849891906928, −8.730863254509022, −7.981986697748899, −7.517318061279521, −7.156151922423359, −6.473675026729157, −6.247872487001163, −5.777555888716403, −5.215335914570350, −4.407200072253991, −4.148549317293154, −3.469970566777644, −2.797184121502179, −2.377299838629507, −1.362121239600153, −1.148897933174536, −0.4345409749811932, 0.4345409749811932, 1.148897933174536, 1.362121239600153, 2.377299838629507, 2.797184121502179, 3.469970566777644, 4.148549317293154, 4.407200072253991, 5.215335914570350, 5.777555888716403, 6.247872487001163, 6.473675026729157, 7.156151922423359, 7.517318061279521, 7.981986697748899, 8.730863254509022, 9.055849891906928, 9.465423549975462, 9.750990547232765, 10.62273250455371, 10.85564192093186, 11.10152720049267, 11.68014515184052, 12.23771573727638, 12.49365460409229

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